Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.8% → 83.1%

Time: 20.0s

Alternatives: 20

Speedup: 3.7×

• 27.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) + 1.0))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 54.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) + 1.0))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 83.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + t_1\right)\right) \leq \infty:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + t_1\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \left(\cos k \cdot \frac{{\sin k}^{-2}}{t}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<=
        (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (+ 1.0 (+ 1.0 t_1)))
        INFINITY)
     (* l (* l (/ 2.0 (* (tan k) (* (+ 2.0 t_1) (* (pow t 3.0) (sin k)))))))
     (* 2.0 (* (/ l k) (* (/ l k) (* (cos k) (/ (pow (sin k) -2.0) t))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if (((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + t_1))) <= ((double) INFINITY)) {
		tmp = l * (l * (2.0 / (tan(k) * ((2.0 + t_1) * (pow(t, 3.0) * sin(k))))));
	} else {
		tmp = 2.0 * ((l / k) * ((l / k) * (cos(k) * (pow(sin(k), -2.0) / t))));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if (((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * (1.0 + (1.0 + t_1))) <= Double.POSITIVE_INFINITY) {
		tmp = l * (l * (2.0 / (Math.tan(k) * ((2.0 + t_1) * (Math.pow(t, 3.0) * Math.sin(k))))));
	} else {
		tmp = 2.0 * ((l / k) * ((l / k) * (Math.cos(k) * (Math.pow(Math.sin(k), -2.0) / t))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow((k / t), 2.0)
	tmp = 0
	if ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * (1.0 + (1.0 + t_1))) <= math.inf:
		tmp = l * (l * (2.0 / (math.tan(k) * ((2.0 + t_1) * (math.pow(t, 3.0) * math.sin(k))))))
	else:
		tmp = 2.0 * ((l / k) * ((l / k) * (math.cos(k) * (math.pow(math.sin(k), -2.0) / t))))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(1.0 + Float64(1.0 + t_1))) <= Inf)
		tmp = Float64(l * Float64(l * Float64(2.0 / Float64(tan(k) * Float64(Float64(2.0 + t_1) * Float64((t ^ 3.0) * sin(k)))))));
	else
		tmp = Float64(2.0 * Float64(Float64(l / k) * Float64(Float64(l / k) * Float64(cos(k) * Float64((sin(k) ^ -2.0) / t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / t) ^ 2.0;
	tmp = 0.0;
	if ((((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + t_1))) <= Inf)
		tmp = l * (l * (2.0 / (tan(k) * ((2.0 + t_1) * ((t ^ 3.0) * sin(k))))));
	else
		tmp = 2.0 * ((l / k) * ((l / k) * (cos(k) * ((sin(k) ^ -2.0) / t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(l * N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[(2.0 + t$95$1), $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], -2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) < +inf.0

   1. Initial program 76.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 76.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 78.1%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 76.9%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 76.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 76.7%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 76.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 76.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 76.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 76.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 76.7%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 76.7%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 60.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)} \]

      2. expm1-udef 55.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)} - 1} \]

      3. associate-*l* 58.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}\right)} - 1 \]

   5. Applied egg-rr 58.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 63.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)\right)} \]

      2. expm1-log1p 80.9%

         \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)} \]

      3. *-commutative 80.9%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{\left(\sin k \cdot {t}^{3}\right)}\right)}\right) \]

   7. Simplified 80.9%

      \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot {t}^{3}\right)\right)}\right)} \]

   if +inf.0 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))

   1. Initial program 0.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 0.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 0.0%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 0.0%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 0.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 0.0%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 0.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 0.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 0.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 0.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 0.0%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 0.0%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 41.4%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 41.4%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 40.1%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 40.1%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 40.1%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 71.5%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   6. Simplified 71.5%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

   7. Taylor expanded in l around 0 41.4%

      \[\leadsto 2 \cdot \color{blue}{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* 40.2%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{{\sin k}^{2} \cdot t}} \]

      2. associate-*r/ 40.2%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot \frac{{\ell}^{2}}{{k}^{2}}}}{{\sin k}^{2} \cdot t} \]

      3. unpow2 40.2%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{\sin k}^{2} \cdot t} \]

      4. unpow2 40.2%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{{\sin k}^{2} \cdot t} \]

      5. times-frac 71.5%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}}{{\sin k}^{2} \cdot t} \]

      6. unpow2 71.5%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}{{\sin k}^{2} \cdot t} \]

      7. *-commutative 71.5%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}}{{\sin k}^{2} \cdot t} \]

      8. *-commutative 71.5%

         \[\leadsto 2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}{\color{blue}{t \cdot {\sin k}^{2}}} \]

      9. times-frac 71.6%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   9. Simplified 71.6%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]
   10. Step-by-step derivation

       1. pow2 71.6%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

       2. frac-times 71.5%

          \[\leadsto 2 \cdot \color{blue}{\frac{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \cos k}{t \cdot {\sin k}^{2}}} \]

       3. associate-*r/ 68.1%

          \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\frac{\ell}{k} \cdot \ell}{k}} \cdot \cos k}{t \cdot {\sin k}^{2}} \]

       4. *-commutative 68.1%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\ell}{k} \cdot \ell}{k} \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot t}} \]

       5. associate-*r/ 68.2%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{\frac{\ell}{k} \cdot \ell}{k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

       6. frac-times 74.2%

          \[\leadsto 2 \cdot \color{blue}{\frac{\left(\frac{\ell}{k} \cdot \ell\right) \cdot \cos k}{k \cdot \left({\sin k}^{2} \cdot t\right)}} \]

       7. *-commutative 74.2%

          \[\leadsto 2 \cdot \frac{\color{blue}{\left(\ell \cdot \frac{\ell}{k}\right)} \cdot \cos k}{k \cdot \left({\sin k}^{2} \cdot t\right)} \]

       8. *-commutative 74.2%

          \[\leadsto 2 \cdot \frac{\left(\ell \cdot \frac{\ell}{k}\right) \cdot \cos k}{k \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

   11. Applied egg-rr 74.2%

       \[\leadsto 2 \cdot \color{blue}{\frac{\left(\ell \cdot \frac{\ell}{k}\right) \cdot \cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   12. Step-by-step derivation

       1. times-frac 68.2%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell \cdot \frac{\ell}{k}}{k} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

       2. associate-*l/ 71.5%

          \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]

       3. *-commutative 71.5%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{{\sin k}^{2} \cdot t}}\right) \]

       4. un-div-inv 71.5%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\cos k \cdot \frac{1}{{\sin k}^{2} \cdot t}\right)}\right) \]

       5. associate-*l* 79.0%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \left(\cos k \cdot \frac{1}{{\sin k}^{2} \cdot t}\right)\right)\right)} \]

       6. associate-/r* 79.0%

          \[\leadsto 2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \left(\cos k \cdot \color{blue}{\frac{\frac{1}{{\sin k}^{2}}}{t}}\right)\right)\right) \]

       7. pow-flip 79.0%

          \[\leadsto 2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \left(\cos k \cdot \frac{\color{blue}{{\sin k}^{\left(-2\right)}}}{t}\right)\right)\right) \]

       8. metadata-eval 79.0%

          \[\leadsto 2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \left(\cos k \cdot \frac{{\sin k}^{\color{blue}{-2}}}{t}\right)\right)\right) \]

   13. Applied egg-rr 79.0%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \left(\cos k \cdot \frac{{\sin k}^{-2}}{t}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \leq \infty:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \left(\cos k \cdot \frac{{\sin k}^{-2}}{t}\right)\right)\right)\\ \end{array} \]

Alternative 2: 68.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.6 \cdot 10^{-25}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \frac{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\ell}{{t}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \left(\cos k \cdot \frac{{\sin k}^{-2}}{t}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 4.6e-25)
   (/
    2.0
    (* (/ k l) (/ (* (tan k) (+ 2.0 (pow (/ k t) 2.0))) (/ l (pow t 3.0)))))
   (* 2.0 (* (/ l k) (* (/ l k) (* (cos k) (/ (pow (sin k) -2.0) t)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.6e-25) {
		tmp = 2.0 / ((k / l) * ((tan(k) * (2.0 + pow((k / t), 2.0))) / (l / pow(t, 3.0))));
	} else {
		tmp = 2.0 * ((l / k) * ((l / k) * (cos(k) * (pow(sin(k), -2.0) / t))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 4.6d-25) then
        tmp = 2.0d0 / ((k / l) * ((tan(k) * (2.0d0 + ((k / t) ** 2.0d0))) / (l / (t ** 3.0d0))))
    else
        tmp = 2.0d0 * ((l / k) * ((l / k) * (cos(k) * ((sin(k) ** (-2.0d0)) / t))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.6e-25) {
		tmp = 2.0 / ((k / l) * ((Math.tan(k) * (2.0 + Math.pow((k / t), 2.0))) / (l / Math.pow(t, 3.0))));
	} else {
		tmp = 2.0 * ((l / k) * ((l / k) * (Math.cos(k) * (Math.pow(Math.sin(k), -2.0) / t))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 4.6e-25:
		tmp = 2.0 / ((k / l) * ((math.tan(k) * (2.0 + math.pow((k / t), 2.0))) / (l / math.pow(t, 3.0))))
	else:
		tmp = 2.0 * ((l / k) * ((l / k) * (math.cos(k) * (math.pow(math.sin(k), -2.0) / t))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 4.6e-25)
		tmp = Float64(2.0 / Float64(Float64(k / l) * Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t) ^ 2.0))) / Float64(l / (t ^ 3.0)))));
	else
		tmp = Float64(2.0 * Float64(Float64(l / k) * Float64(Float64(l / k) * Float64(cos(k) * Float64((sin(k) ^ -2.0) / t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 4.6e-25)
		tmp = 2.0 / ((k / l) * ((tan(k) * (2.0 + ((k / t) ^ 2.0))) / (l / (t ^ 3.0))));
	else
		tmp = 2.0 * ((l / k) * ((l / k) * (cos(k) * ((sin(k) ^ -2.0) / t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 4.6e-25], N[(2.0 / N[(N[(k / l), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], -2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.5999999999999998e-25

   1. Initial program 52.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 52.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. +-commutative 52.3%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 52.3%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

   4. Taylor expanded in k around 0 52.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. associate-/l* 51.1%

         \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      2. unpow2 51.1%

         \[\leadsto \frac{2}{\frac{k}{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      3. associate-/l* 56.5%

         \[\leadsto \frac{2}{\frac{k}{\color{blue}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   6. Simplified 56.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 49.0%

         \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}} \]

      2. associate-+r+ 49.0%

         \[\leadsto \frac{2}{\frac{k \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}} \]

      3. metadata-eval 49.0%

         \[\leadsto \frac{2}{\frac{k \cdot \left(\tan k \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}} \]

      4. div-inv 49.0%

         \[\leadsto \frac{2}{\frac{k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\color{blue}{\ell \cdot \frac{1}{\frac{{t}^{3}}{\ell}}}}} \]

      5. clear-num 49.0%

         \[\leadsto \frac{2}{\frac{k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\ell \cdot \color{blue}{\frac{\ell}{{t}^{3}}}}} \]

   8. Applied egg-rr 49.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\ell \cdot \frac{\ell}{{t}^{3}}}}} \]
   9. Step-by-step derivation

      1. times-frac 59.2%

         \[\leadsto \frac{2}{\color{blue}{\frac{k}{\ell} \cdot \frac{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\ell}{{t}^{3}}}}} \]

   10. Simplified 59.2%

       \[\leadsto \frac{2}{\color{blue}{\frac{k}{\ell} \cdot \frac{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\ell}{{t}^{3}}}}} \]

   if 4.5999999999999998e-25 < k

   1. Initial program 43.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 43.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 43.4%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 43.4%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 43.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 43.5%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 43.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 43.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 43.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 43.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 43.4%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 43.4%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 71.5%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 71.5%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 69.9%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 69.9%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 69.9%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 86.4%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   6. Simplified 86.4%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

   7. Taylor expanded in l around 0 71.5%

      \[\leadsto 2 \cdot \color{blue}{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* 70.0%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{{\sin k}^{2} \cdot t}} \]

      2. associate-*r/ 70.0%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot \frac{{\ell}^{2}}{{k}^{2}}}}{{\sin k}^{2} \cdot t} \]

      3. unpow2 70.0%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{\sin k}^{2} \cdot t} \]

      4. unpow2 70.0%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{{\sin k}^{2} \cdot t} \]

      5. times-frac 86.3%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}}{{\sin k}^{2} \cdot t} \]

      6. unpow2 86.3%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}{{\sin k}^{2} \cdot t} \]

      7. *-commutative 86.3%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}}{{\sin k}^{2} \cdot t} \]

      8. *-commutative 86.3%

         \[\leadsto 2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}{\color{blue}{t \cdot {\sin k}^{2}}} \]

      9. times-frac 86.4%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   9. Simplified 86.4%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]
   10. Step-by-step derivation

       1. pow2 86.4%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

       2. frac-times 86.3%

          \[\leadsto 2 \cdot \color{blue}{\frac{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \cos k}{t \cdot {\sin k}^{2}}} \]

       3. associate-*r/ 83.7%

          \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\frac{\ell}{k} \cdot \ell}{k}} \cdot \cos k}{t \cdot {\sin k}^{2}} \]

       4. *-commutative 83.7%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\ell}{k} \cdot \ell}{k} \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot t}} \]

       5. associate-*r/ 83.8%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{\frac{\ell}{k} \cdot \ell}{k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

       6. frac-times 87.3%

          \[\leadsto 2 \cdot \color{blue}{\frac{\left(\frac{\ell}{k} \cdot \ell\right) \cdot \cos k}{k \cdot \left({\sin k}^{2} \cdot t\right)}} \]

       7. *-commutative 87.3%

          \[\leadsto 2 \cdot \frac{\color{blue}{\left(\ell \cdot \frac{\ell}{k}\right)} \cdot \cos k}{k \cdot \left({\sin k}^{2} \cdot t\right)} \]

       8. *-commutative 87.3%

          \[\leadsto 2 \cdot \frac{\left(\ell \cdot \frac{\ell}{k}\right) \cdot \cos k}{k \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

   11. Applied egg-rr 87.3%

       \[\leadsto 2 \cdot \color{blue}{\frac{\left(\ell \cdot \frac{\ell}{k}\right) \cdot \cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   12. Step-by-step derivation

       1. times-frac 83.8%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell \cdot \frac{\ell}{k}}{k} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

       2. associate-*l/ 86.4%

          \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]

       3. *-commutative 86.4%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{{\sin k}^{2} \cdot t}}\right) \]

       4. un-div-inv 86.4%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\cos k \cdot \frac{1}{{\sin k}^{2} \cdot t}\right)}\right) \]

       5. associate-*l* 91.7%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \left(\cos k \cdot \frac{1}{{\sin k}^{2} \cdot t}\right)\right)\right)} \]

       6. associate-/r* 91.6%

          \[\leadsto 2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \left(\cos k \cdot \color{blue}{\frac{\frac{1}{{\sin k}^{2}}}{t}}\right)\right)\right) \]

       7. pow-flip 91.6%

          \[\leadsto 2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \left(\cos k \cdot \frac{\color{blue}{{\sin k}^{\left(-2\right)}}}{t}\right)\right)\right) \]

       8. metadata-eval 91.6%

          \[\leadsto 2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \left(\cos k \cdot \frac{{\sin k}^{\color{blue}{-2}}}{t}\right)\right)\right) \]

   13. Applied egg-rr 91.6%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \left(\cos k \cdot \frac{{\sin k}^{-2}}{t}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.6 \cdot 10^{-25}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \frac{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\ell}{{t}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \left(\cos k \cdot \frac{{\sin k}^{-2}}{t}\right)\right)\right)\\ \end{array} \]

Alternative 3: 68.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{k \cdot \left(\frac{k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.45e-23)
   (/ 1.0 (* k (* (/ k l) (/ (pow t 3.0) l))))
   (* 2.0 (* (* (/ l k) (/ l k)) (/ (cos k) (* t (pow (sin k) 2.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.45e-23) {
		tmp = 1.0 / (k * ((k / l) * (pow(t, 3.0) / l)));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * pow(sin(k), 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.45d-23) then
        tmp = 1.0d0 / (k * ((k / l) * ((t ** 3.0d0) / l)))
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) * (cos(k) / (t * (sin(k) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.45e-23) {
		tmp = 1.0 / (k * ((k / l) * (Math.pow(t, 3.0) / l)));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (Math.cos(k) / (t * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.45e-23:
		tmp = 1.0 / (k * ((k / l) * (math.pow(t, 3.0) / l)))
	else:
		tmp = 2.0 * (((l / k) * (l / k)) * (math.cos(k) / (t * math.pow(math.sin(k), 2.0))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.45e-23)
		tmp = Float64(1.0 / Float64(k * Float64(Float64(k / l) * Float64((t ^ 3.0) / l))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.45e-23)
		tmp = 1.0 / (k * ((k / l) * ((t ^ 3.0) / l)));
	else
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * (sin(k) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.45e-23], N[(1.0 / N[(k * N[(N[(k / l), $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.4500000000000001e-23

   1. Initial program 52.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 52.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. +-commutative 52.3%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 52.3%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

   4. Taylor expanded in k around 0 46.4%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-/l* 45.9%

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{{t}^{3}}}}} \]

      2. unpow2 45.9%

         \[\leadsto \frac{2}{2 \cdot \frac{\color{blue}{k \cdot k}}{\frac{{\ell}^{2}}{{t}^{3}}}} \]

      3. unpow2 45.9%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}}} \]

      4. associate-/l* 51.3%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}} \]

   6. Simplified 51.3%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{k \cdot k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 36.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{2 \cdot \frac{k \cdot k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}\right)\right)} \]

      2. expm1-udef 35.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{2 \cdot \frac{k \cdot k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}\right)} - 1} \]

      3. associate-/r* 35.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{2}{2}}{\frac{k \cdot k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}}\right)} - 1 \]

      4. metadata-eval 35.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\frac{k \cdot k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}\right)} - 1 \]

      5. associate-/r/ 34.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\frac{k \cdot k}{\ell} \cdot \frac{{t}^{3}}{\ell}}}\right)} - 1 \]

   8. Applied egg-rr 34.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\frac{k \cdot k}{\ell} \cdot \frac{{t}^{3}}{\ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 37.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{k \cdot k}{\ell} \cdot \frac{{t}^{3}}{\ell}}\right)\right)} \]

      2. expm1-log1p 51.1%

         \[\leadsto \color{blue}{\frac{1}{\frac{k \cdot k}{\ell} \cdot \frac{{t}^{3}}{\ell}}} \]

      3. unpow2 51.1%

         \[\leadsto \frac{1}{\frac{\color{blue}{{k}^{2}}}{\ell} \cdot \frac{{t}^{3}}{\ell}} \]

      4. times-frac 46.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{{k}^{2} \cdot {t}^{3}}{\ell \cdot \ell}}} \]

      5. unpow2 46.4%

         \[\leadsto \frac{1}{\frac{{k}^{2} \cdot {t}^{3}}{\color{blue}{{\ell}^{2}}}} \]

      6. associate-/l* 45.9%

         \[\leadsto \frac{1}{\color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{{t}^{3}}}}} \]

      7. unpow2 45.9%

         \[\leadsto \frac{1}{\frac{{k}^{2}}{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}}} \]

      8. associate-*r/ 51.3%

         \[\leadsto \frac{1}{\frac{{k}^{2}}{\color{blue}{\ell \cdot \frac{\ell}{{t}^{3}}}}} \]

      9. unpow2 51.3%

         \[\leadsto \frac{1}{\frac{\color{blue}{k \cdot k}}{\ell \cdot \frac{\ell}{{t}^{3}}}} \]

      10. associate-*r/ 58.9%

          \[\leadsto \frac{1}{\color{blue}{k \cdot \frac{k}{\ell \cdot \frac{\ell}{{t}^{3}}}}} \]

      11. associate-*r/ 53.5%

          \[\leadsto \frac{1}{k \cdot \frac{k}{\color{blue}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]

      12. unpow2 53.5%

          \[\leadsto \frac{1}{k \cdot \frac{k}{\frac{\color{blue}{{\ell}^{2}}}{{t}^{3}}}} \]

      13. associate-/l* 54.7%

          \[\leadsto \frac{1}{k \cdot \color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}}} \]

      14. unpow2 54.7%

          \[\leadsto \frac{1}{k \cdot \frac{k \cdot {t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      15. times-frac 60.0%

          \[\leadsto \frac{1}{k \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)}} \]

      16. *-commutative 60.0%

          \[\leadsto \frac{1}{k \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right)}} \]

   10. Simplified 60.0%

       \[\leadsto \color{blue}{\frac{1}{k \cdot \left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right)}} \]

   if 1.4500000000000001e-23 < k

   1. Initial program 43.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 43.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 43.4%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 43.4%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 43.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 43.5%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 43.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 43.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 43.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 43.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 43.4%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 43.4%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 71.5%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 71.5%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 69.9%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 69.9%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 69.9%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 86.4%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   6. Simplified 86.4%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{k \cdot \left(\frac{k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 4: 69.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.9 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{k \cdot \left(\frac{k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \left(\cos k \cdot \frac{{\sin k}^{-2}}{t}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.9e-24)
   (/ 1.0 (* k (* (/ k l) (/ (pow t 3.0) l))))
   (* 2.0 (* (/ l k) (* (/ l k) (* (cos k) (/ (pow (sin k) -2.0) t)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.9e-24) {
		tmp = 1.0 / (k * ((k / l) * (pow(t, 3.0) / l)));
	} else {
		tmp = 2.0 * ((l / k) * ((l / k) * (cos(k) * (pow(sin(k), -2.0) / t))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.9d-24) then
        tmp = 1.0d0 / (k * ((k / l) * ((t ** 3.0d0) / l)))
    else
        tmp = 2.0d0 * ((l / k) * ((l / k) * (cos(k) * ((sin(k) ** (-2.0d0)) / t))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.9e-24) {
		tmp = 1.0 / (k * ((k / l) * (Math.pow(t, 3.0) / l)));
	} else {
		tmp = 2.0 * ((l / k) * ((l / k) * (Math.cos(k) * (Math.pow(Math.sin(k), -2.0) / t))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.9e-24:
		tmp = 1.0 / (k * ((k / l) * (math.pow(t, 3.0) / l)))
	else:
		tmp = 2.0 * ((l / k) * ((l / k) * (math.cos(k) * (math.pow(math.sin(k), -2.0) / t))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.9e-24)
		tmp = Float64(1.0 / Float64(k * Float64(Float64(k / l) * Float64((t ^ 3.0) / l))));
	else
		tmp = Float64(2.0 * Float64(Float64(l / k) * Float64(Float64(l / k) * Float64(cos(k) * Float64((sin(k) ^ -2.0) / t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.9e-24)
		tmp = 1.0 / (k * ((k / l) * ((t ^ 3.0) / l)));
	else
		tmp = 2.0 * ((l / k) * ((l / k) * (cos(k) * ((sin(k) ^ -2.0) / t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.9e-24], N[(1.0 / N[(k * N[(N[(k / l), $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], -2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.8999999999999999e-24

   1. Initial program 52.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 52.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. +-commutative 52.3%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 52.3%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

   4. Taylor expanded in k around 0 46.4%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-/l* 45.9%

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{{t}^{3}}}}} \]

      2. unpow2 45.9%

         \[\leadsto \frac{2}{2 \cdot \frac{\color{blue}{k \cdot k}}{\frac{{\ell}^{2}}{{t}^{3}}}} \]

      3. unpow2 45.9%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}}} \]

      4. associate-/l* 51.3%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}} \]

   6. Simplified 51.3%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{k \cdot k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 36.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{2 \cdot \frac{k \cdot k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}\right)\right)} \]

      2. expm1-udef 35.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{2 \cdot \frac{k \cdot k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}\right)} - 1} \]

      3. associate-/r* 35.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{2}{2}}{\frac{k \cdot k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}}\right)} - 1 \]

      4. metadata-eval 35.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\frac{k \cdot k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}\right)} - 1 \]

      5. associate-/r/ 34.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\frac{k \cdot k}{\ell} \cdot \frac{{t}^{3}}{\ell}}}\right)} - 1 \]

   8. Applied egg-rr 34.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\frac{k \cdot k}{\ell} \cdot \frac{{t}^{3}}{\ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 37.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{k \cdot k}{\ell} \cdot \frac{{t}^{3}}{\ell}}\right)\right)} \]

      2. expm1-log1p 51.1%

         \[\leadsto \color{blue}{\frac{1}{\frac{k \cdot k}{\ell} \cdot \frac{{t}^{3}}{\ell}}} \]

      3. unpow2 51.1%

         \[\leadsto \frac{1}{\frac{\color{blue}{{k}^{2}}}{\ell} \cdot \frac{{t}^{3}}{\ell}} \]

      4. times-frac 46.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{{k}^{2} \cdot {t}^{3}}{\ell \cdot \ell}}} \]

      5. unpow2 46.4%

         \[\leadsto \frac{1}{\frac{{k}^{2} \cdot {t}^{3}}{\color{blue}{{\ell}^{2}}}} \]

      6. associate-/l* 45.9%

         \[\leadsto \frac{1}{\color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{{t}^{3}}}}} \]

      7. unpow2 45.9%

         \[\leadsto \frac{1}{\frac{{k}^{2}}{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}}} \]

      8. associate-*r/ 51.3%

         \[\leadsto \frac{1}{\frac{{k}^{2}}{\color{blue}{\ell \cdot \frac{\ell}{{t}^{3}}}}} \]

      9. unpow2 51.3%

         \[\leadsto \frac{1}{\frac{\color{blue}{k \cdot k}}{\ell \cdot \frac{\ell}{{t}^{3}}}} \]

      10. associate-*r/ 58.9%

          \[\leadsto \frac{1}{\color{blue}{k \cdot \frac{k}{\ell \cdot \frac{\ell}{{t}^{3}}}}} \]

      11. associate-*r/ 53.5%

          \[\leadsto \frac{1}{k \cdot \frac{k}{\color{blue}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]

      12. unpow2 53.5%

          \[\leadsto \frac{1}{k \cdot \frac{k}{\frac{\color{blue}{{\ell}^{2}}}{{t}^{3}}}} \]

      13. associate-/l* 54.7%

          \[\leadsto \frac{1}{k \cdot \color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}}} \]

      14. unpow2 54.7%

          \[\leadsto \frac{1}{k \cdot \frac{k \cdot {t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      15. times-frac 60.0%

          \[\leadsto \frac{1}{k \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)}} \]

      16. *-commutative 60.0%

          \[\leadsto \frac{1}{k \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right)}} \]

   10. Simplified 60.0%

       \[\leadsto \color{blue}{\frac{1}{k \cdot \left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right)}} \]

   if 2.8999999999999999e-24 < k

   1. Initial program 43.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 43.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 43.4%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 43.4%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 43.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 43.5%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 43.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 43.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 43.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 43.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 43.4%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 43.4%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 71.5%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 71.5%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 69.9%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 69.9%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 69.9%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 86.4%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   6. Simplified 86.4%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

   7. Taylor expanded in l around 0 71.5%

      \[\leadsto 2 \cdot \color{blue}{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* 70.0%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{{\sin k}^{2} \cdot t}} \]

      2. associate-*r/ 70.0%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot \frac{{\ell}^{2}}{{k}^{2}}}}{{\sin k}^{2} \cdot t} \]

      3. unpow2 70.0%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{\sin k}^{2} \cdot t} \]

      4. unpow2 70.0%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{{\sin k}^{2} \cdot t} \]

      5. times-frac 86.3%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}}{{\sin k}^{2} \cdot t} \]

      6. unpow2 86.3%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}{{\sin k}^{2} \cdot t} \]

      7. *-commutative 86.3%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}}{{\sin k}^{2} \cdot t} \]

      8. *-commutative 86.3%

         \[\leadsto 2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}{\color{blue}{t \cdot {\sin k}^{2}}} \]

      9. times-frac 86.4%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   9. Simplified 86.4%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]
   10. Step-by-step derivation

       1. pow2 86.4%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

       2. frac-times 86.3%

          \[\leadsto 2 \cdot \color{blue}{\frac{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \cos k}{t \cdot {\sin k}^{2}}} \]

       3. associate-*r/ 83.7%

          \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\frac{\ell}{k} \cdot \ell}{k}} \cdot \cos k}{t \cdot {\sin k}^{2}} \]

       4. *-commutative 83.7%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\ell}{k} \cdot \ell}{k} \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot t}} \]

       5. associate-*r/ 83.8%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{\frac{\ell}{k} \cdot \ell}{k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

       6. frac-times 87.3%

          \[\leadsto 2 \cdot \color{blue}{\frac{\left(\frac{\ell}{k} \cdot \ell\right) \cdot \cos k}{k \cdot \left({\sin k}^{2} \cdot t\right)}} \]

       7. *-commutative 87.3%

          \[\leadsto 2 \cdot \frac{\color{blue}{\left(\ell \cdot \frac{\ell}{k}\right)} \cdot \cos k}{k \cdot \left({\sin k}^{2} \cdot t\right)} \]

       8. *-commutative 87.3%

          \[\leadsto 2 \cdot \frac{\left(\ell \cdot \frac{\ell}{k}\right) \cdot \cos k}{k \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

   11. Applied egg-rr 87.3%

       \[\leadsto 2 \cdot \color{blue}{\frac{\left(\ell \cdot \frac{\ell}{k}\right) \cdot \cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   12. Step-by-step derivation

       1. times-frac 83.8%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell \cdot \frac{\ell}{k}}{k} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

       2. associate-*l/ 86.4%

          \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]

       3. *-commutative 86.4%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{{\sin k}^{2} \cdot t}}\right) \]

       4. un-div-inv 86.4%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\cos k \cdot \frac{1}{{\sin k}^{2} \cdot t}\right)}\right) \]

       5. associate-*l* 91.7%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \left(\cos k \cdot \frac{1}{{\sin k}^{2} \cdot t}\right)\right)\right)} \]

       6. associate-/r* 91.6%

          \[\leadsto 2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \left(\cos k \cdot \color{blue}{\frac{\frac{1}{{\sin k}^{2}}}{t}}\right)\right)\right) \]

       7. pow-flip 91.6%

          \[\leadsto 2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \left(\cos k \cdot \frac{\color{blue}{{\sin k}^{\left(-2\right)}}}{t}\right)\right)\right) \]

       8. metadata-eval 91.6%

          \[\leadsto 2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \left(\cos k \cdot \frac{{\sin k}^{\color{blue}{-2}}}{t}\right)\right)\right) \]

   13. Applied egg-rr 91.6%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \left(\cos k \cdot \frac{{\sin k}^{-2}}{t}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.9 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{k \cdot \left(\frac{k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \left(\cos k \cdot \frac{{\sin k}^{-2}}{t}\right)\right)\right)\\ \end{array} \]

Alternative 5: 68.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{k \cdot \left(\frac{k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \frac{\ell}{k}\right)}{k \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 9.5e-10)
   (/ 1.0 (* k (* (/ k l) (/ (pow t 3.0) l))))
   (*
    2.0
    (/ (* (cos k) (* l (/ l k))) (* k (* t (- 0.5 (/ (cos (+ k k)) 2.0))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 9.5e-10) {
		tmp = 1.0 / (k * ((k / l) * (pow(t, 3.0) / l)));
	} else {
		tmp = 2.0 * ((cos(k) * (l * (l / k))) / (k * (t * (0.5 - (cos((k + k)) / 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 9.5d-10) then
        tmp = 1.0d0 / (k * ((k / l) * ((t ** 3.0d0) / l)))
    else
        tmp = 2.0d0 * ((cos(k) * (l * (l / k))) / (k * (t * (0.5d0 - (cos((k + k)) / 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 9.5e-10) {
		tmp = 1.0 / (k * ((k / l) * (Math.pow(t, 3.0) / l)));
	} else {
		tmp = 2.0 * ((Math.cos(k) * (l * (l / k))) / (k * (t * (0.5 - (Math.cos((k + k)) / 2.0)))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 9.5e-10:
		tmp = 1.0 / (k * ((k / l) * (math.pow(t, 3.0) / l)))
	else:
		tmp = 2.0 * ((math.cos(k) * (l * (l / k))) / (k * (t * (0.5 - (math.cos((k + k)) / 2.0)))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 9.5e-10)
		tmp = Float64(1.0 / Float64(k * Float64(Float64(k / l) * Float64((t ^ 3.0) / l))));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) * Float64(l * Float64(l / k))) / Float64(k * Float64(t * Float64(0.5 - Float64(cos(Float64(k + k)) / 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 9.5e-10)
		tmp = 1.0 / (k * ((k / l) * ((t ^ 3.0) / l)));
	else
		tmp = 2.0 * ((cos(k) * (l * (l / k))) / (k * (t * (0.5 - (cos((k + k)) / 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 9.5e-10], N[(1.0 / N[(k * N[(N[(k / l), $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[(l * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * N[(t * N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.50000000000000028e-10

   1. Initial program 51.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 51.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. +-commutative 51.5%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 51.5%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

   4. Taylor expanded in k around 0 45.7%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-/l* 45.2%

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{{t}^{3}}}}} \]

      2. unpow2 45.2%

         \[\leadsto \frac{2}{2 \cdot \frac{\color{blue}{k \cdot k}}{\frac{{\ell}^{2}}{{t}^{3}}}} \]

      3. unpow2 45.2%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}}} \]

      4. associate-/l* 50.5%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}} \]

   6. Simplified 50.5%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{k \cdot k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 36.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{2 \cdot \frac{k \cdot k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}\right)\right)} \]

      2. expm1-udef 34.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{2 \cdot \frac{k \cdot k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}\right)} - 1} \]

      3. associate-/r* 34.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{2}{2}}{\frac{k \cdot k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}}\right)} - 1 \]

      4. metadata-eval 34.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\frac{k \cdot k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}\right)} - 1 \]

      5. associate-/r/ 34.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\frac{k \cdot k}{\ell} \cdot \frac{{t}^{3}}{\ell}}}\right)} - 1 \]

   8. Applied egg-rr 34.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\frac{k \cdot k}{\ell} \cdot \frac{{t}^{3}}{\ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 36.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{k \cdot k}{\ell} \cdot \frac{{t}^{3}}{\ell}}\right)\right)} \]

      2. expm1-log1p 50.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{k \cdot k}{\ell} \cdot \frac{{t}^{3}}{\ell}}} \]

      3. unpow2 50.3%

         \[\leadsto \frac{1}{\frac{\color{blue}{{k}^{2}}}{\ell} \cdot \frac{{t}^{3}}{\ell}} \]

      4. times-frac 45.7%

         \[\leadsto \frac{1}{\color{blue}{\frac{{k}^{2} \cdot {t}^{3}}{\ell \cdot \ell}}} \]

      5. unpow2 45.7%

         \[\leadsto \frac{1}{\frac{{k}^{2} \cdot {t}^{3}}{\color{blue}{{\ell}^{2}}}} \]

      6. associate-/l* 45.2%

         \[\leadsto \frac{1}{\color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{{t}^{3}}}}} \]

      7. unpow2 45.2%

         \[\leadsto \frac{1}{\frac{{k}^{2}}{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}}} \]

      8. associate-*r/ 50.5%

         \[\leadsto \frac{1}{\frac{{k}^{2}}{\color{blue}{\ell \cdot \frac{\ell}{{t}^{3}}}}} \]

      9. unpow2 50.5%

         \[\leadsto \frac{1}{\frac{\color{blue}{k \cdot k}}{\ell \cdot \frac{\ell}{{t}^{3}}}} \]

      10. associate-*r/ 58.1%

          \[\leadsto \frac{1}{\color{blue}{k \cdot \frac{k}{\ell \cdot \frac{\ell}{{t}^{3}}}}} \]

      11. associate-*r/ 52.7%

          \[\leadsto \frac{1}{k \cdot \frac{k}{\color{blue}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]

      12. unpow2 52.7%

          \[\leadsto \frac{1}{k \cdot \frac{k}{\frac{\color{blue}{{\ell}^{2}}}{{t}^{3}}}} \]

      13. associate-/l* 53.9%

          \[\leadsto \frac{1}{k \cdot \color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}}} \]

      14. unpow2 53.9%

          \[\leadsto \frac{1}{k \cdot \frac{k \cdot {t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      15. times-frac 59.1%

          \[\leadsto \frac{1}{k \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)}} \]

      16. *-commutative 59.1%

          \[\leadsto \frac{1}{k \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right)}} \]

   10. Simplified 59.1%

       \[\leadsto \color{blue}{\frac{1}{k \cdot \left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right)}} \]

   if 9.50000000000000028e-10 < k

   1. Initial program 45.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 45.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 45.2%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 45.2%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 45.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 45.3%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 45.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 45.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 45.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 45.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 45.2%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 45.2%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 71.7%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 71.7%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 70.1%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 70.1%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 70.1%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 87.3%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   6. Simplified 87.3%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

   7. Taylor expanded in l around 0 71.7%

      \[\leadsto 2 \cdot \color{blue}{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* 70.1%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{{\sin k}^{2} \cdot t}} \]

      2. associate-*r/ 70.1%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot \frac{{\ell}^{2}}{{k}^{2}}}}{{\sin k}^{2} \cdot t} \]

      3. unpow2 70.1%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{\sin k}^{2} \cdot t} \]

      4. unpow2 70.1%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{{\sin k}^{2} \cdot t} \]

      5. times-frac 87.2%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}}{{\sin k}^{2} \cdot t} \]

      6. unpow2 87.2%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}{{\sin k}^{2} \cdot t} \]

      7. *-commutative 87.2%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}}{{\sin k}^{2} \cdot t} \]

      8. *-commutative 87.2%

         \[\leadsto 2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}{\color{blue}{t \cdot {\sin k}^{2}}} \]

      9. times-frac 87.3%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   9. Simplified 87.3%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]
   10. Step-by-step derivation

       1. pow2 87.3%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

       2. frac-times 87.2%

          \[\leadsto 2 \cdot \color{blue}{\frac{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \cos k}{t \cdot {\sin k}^{2}}} \]

       3. associate-*r/ 84.4%

          \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\frac{\ell}{k} \cdot \ell}{k}} \cdot \cos k}{t \cdot {\sin k}^{2}} \]

       4. *-commutative 84.4%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\ell}{k} \cdot \ell}{k} \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot t}} \]

       5. associate-*r/ 84.5%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{\frac{\ell}{k} \cdot \ell}{k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

       6. frac-times 88.3%

          \[\leadsto 2 \cdot \color{blue}{\frac{\left(\frac{\ell}{k} \cdot \ell\right) \cdot \cos k}{k \cdot \left({\sin k}^{2} \cdot t\right)}} \]

       7. *-commutative 88.3%

          \[\leadsto 2 \cdot \frac{\color{blue}{\left(\ell \cdot \frac{\ell}{k}\right)} \cdot \cos k}{k \cdot \left({\sin k}^{2} \cdot t\right)} \]

       8. *-commutative 88.3%

          \[\leadsto 2 \cdot \frac{\left(\ell \cdot \frac{\ell}{k}\right) \cdot \cos k}{k \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

   11. Applied egg-rr 88.3%

       \[\leadsto 2 \cdot \color{blue}{\frac{\left(\ell \cdot \frac{\ell}{k}\right) \cdot \cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   12. Step-by-step derivation

       1. unpow2 88.3%

          \[\leadsto 2 \cdot \frac{\left(\ell \cdot \frac{\ell}{k}\right) \cdot \cos k}{k \cdot \left(t \cdot \color{blue}{\left(\sin k \cdot \sin k\right)}\right)} \]

       2. sin-mult 88.1%

          \[\leadsto 2 \cdot \frac{\left(\ell \cdot \frac{\ell}{k}\right) \cdot \cos k}{k \cdot \left(t \cdot \color{blue}{\frac{\cos \left(k - k\right) - \cos \left(k + k\right)}{2}}\right)} \]

   13. Applied egg-rr 88.1%

       \[\leadsto 2 \cdot \frac{\left(\ell \cdot \frac{\ell}{k}\right) \cdot \cos k}{k \cdot \left(t \cdot \color{blue}{\frac{\cos \left(k - k\right) - \cos \left(k + k\right)}{2}}\right)} \]
   14. Step-by-step derivation

       1. div-sub 88.1%

          \[\leadsto 2 \cdot \frac{\left(\ell \cdot \frac{\ell}{k}\right) \cdot \cos k}{k \cdot \left(t \cdot \color{blue}{\left(\frac{\cos \left(k - k\right)}{2} - \frac{\cos \left(k + k\right)}{2}\right)}\right)} \]

       2. +-inverses 88.1%

          \[\leadsto 2 \cdot \frac{\left(\ell \cdot \frac{\ell}{k}\right) \cdot \cos k}{k \cdot \left(t \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(k + k\right)}{2}\right)\right)} \]

       3. cos-0 88.1%

          \[\leadsto 2 \cdot \frac{\left(\ell \cdot \frac{\ell}{k}\right) \cdot \cos k}{k \cdot \left(t \cdot \left(\frac{\color{blue}{1}}{2} - \frac{\cos \left(k + k\right)}{2}\right)\right)} \]

       4. metadata-eval 88.1%

          \[\leadsto 2 \cdot \frac{\left(\ell \cdot \frac{\ell}{k}\right) \cdot \cos k}{k \cdot \left(t \cdot \left(\color{blue}{0.5} - \frac{\cos \left(k + k\right)}{2}\right)\right)} \]

   15. Simplified 88.1%

       \[\leadsto 2 \cdot \frac{\left(\ell \cdot \frac{\ell}{k}\right) \cdot \cos k}{k \cdot \left(t \cdot \color{blue}{\left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{k \cdot \left(\frac{k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \frac{\ell}{k}\right)}{k \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)\right)}\\ \end{array} \]

Alternative 6: 65.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{k \cdot \left(\frac{k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.5e-23)
   (/ 1.0 (* k (* (/ k l) (/ (pow t 3.0) l))))
   (* (* l l) (/ 2.0 (* (tan k) (* k (* k (* t (sin k)))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.5e-23) {
		tmp = 1.0 / (k * ((k / l) * (pow(t, 3.0) / l)));
	} else {
		tmp = (l * l) * (2.0 / (tan(k) * (k * (k * (t * sin(k))))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.5d-23) then
        tmp = 1.0d0 / (k * ((k / l) * ((t ** 3.0d0) / l)))
    else
        tmp = (l * l) * (2.0d0 / (tan(k) * (k * (k * (t * sin(k))))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.5e-23) {
		tmp = 1.0 / (k * ((k / l) * (Math.pow(t, 3.0) / l)));
	} else {
		tmp = (l * l) * (2.0 / (Math.tan(k) * (k * (k * (t * Math.sin(k))))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.5e-23:
		tmp = 1.0 / (k * ((k / l) * (math.pow(t, 3.0) / l)))
	else:
		tmp = (l * l) * (2.0 / (math.tan(k) * (k * (k * (t * math.sin(k))))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.5e-23)
		tmp = Float64(1.0 / Float64(k * Float64(Float64(k / l) * Float64((t ^ 3.0) / l))));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(tan(k) * Float64(k * Float64(k * Float64(t * sin(k)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.5e-23)
		tmp = 1.0 / (k * ((k / l) * ((t ^ 3.0) / l)));
	else
		tmp = (l * l) * (2.0 / (tan(k) * (k * (k * (t * sin(k))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.5e-23], N[(1.0 / N[(k * N[(N[(k / l), $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k * N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.5000000000000001e-23

   1. Initial program 52.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 52.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. +-commutative 52.3%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 52.3%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

   4. Taylor expanded in k around 0 46.4%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-/l* 45.9%

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{{t}^{3}}}}} \]

      2. unpow2 45.9%

         \[\leadsto \frac{2}{2 \cdot \frac{\color{blue}{k \cdot k}}{\frac{{\ell}^{2}}{{t}^{3}}}} \]

      3. unpow2 45.9%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}}} \]

      4. associate-/l* 51.3%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}} \]

   6. Simplified 51.3%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{k \cdot k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 36.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{2 \cdot \frac{k \cdot k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}\right)\right)} \]

      2. expm1-udef 35.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{2 \cdot \frac{k \cdot k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}\right)} - 1} \]

      3. associate-/r* 35.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{2}{2}}{\frac{k \cdot k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}}\right)} - 1 \]

      4. metadata-eval 35.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\frac{k \cdot k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}\right)} - 1 \]

      5. associate-/r/ 34.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\frac{k \cdot k}{\ell} \cdot \frac{{t}^{3}}{\ell}}}\right)} - 1 \]

   8. Applied egg-rr 34.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\frac{k \cdot k}{\ell} \cdot \frac{{t}^{3}}{\ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 37.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{k \cdot k}{\ell} \cdot \frac{{t}^{3}}{\ell}}\right)\right)} \]

      2. expm1-log1p 51.1%

         \[\leadsto \color{blue}{\frac{1}{\frac{k \cdot k}{\ell} \cdot \frac{{t}^{3}}{\ell}}} \]

      3. unpow2 51.1%

         \[\leadsto \frac{1}{\frac{\color{blue}{{k}^{2}}}{\ell} \cdot \frac{{t}^{3}}{\ell}} \]

      4. times-frac 46.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{{k}^{2} \cdot {t}^{3}}{\ell \cdot \ell}}} \]

      5. unpow2 46.4%

         \[\leadsto \frac{1}{\frac{{k}^{2} \cdot {t}^{3}}{\color{blue}{{\ell}^{2}}}} \]

      6. associate-/l* 45.9%

         \[\leadsto \frac{1}{\color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{{t}^{3}}}}} \]

      7. unpow2 45.9%

         \[\leadsto \frac{1}{\frac{{k}^{2}}{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}}} \]

      8. associate-*r/ 51.3%

         \[\leadsto \frac{1}{\frac{{k}^{2}}{\color{blue}{\ell \cdot \frac{\ell}{{t}^{3}}}}} \]

      9. unpow2 51.3%

         \[\leadsto \frac{1}{\frac{\color{blue}{k \cdot k}}{\ell \cdot \frac{\ell}{{t}^{3}}}} \]

      10. associate-*r/ 58.9%

          \[\leadsto \frac{1}{\color{blue}{k \cdot \frac{k}{\ell \cdot \frac{\ell}{{t}^{3}}}}} \]

      11. associate-*r/ 53.5%

          \[\leadsto \frac{1}{k \cdot \frac{k}{\color{blue}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]

      12. unpow2 53.5%

          \[\leadsto \frac{1}{k \cdot \frac{k}{\frac{\color{blue}{{\ell}^{2}}}{{t}^{3}}}} \]

      13. associate-/l* 54.7%

          \[\leadsto \frac{1}{k \cdot \color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}}} \]

      14. unpow2 54.7%

          \[\leadsto \frac{1}{k \cdot \frac{k \cdot {t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      15. times-frac 60.0%

          \[\leadsto \frac{1}{k \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)}} \]

      16. *-commutative 60.0%

          \[\leadsto \frac{1}{k \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right)}} \]

   10. Simplified 60.0%

       \[\leadsto \color{blue}{\frac{1}{k \cdot \left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right)}} \]

   if 2.5000000000000001e-23 < k

   1. Initial program 43.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 43.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 43.4%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 43.4%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 43.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 43.5%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 43.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 43.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 43.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 43.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 43.4%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 43.4%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 71.4%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. unpow2 71.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(\sin k \cdot t\right)\right)} \]

      2. associate-*l* 78.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(k \cdot \left(k \cdot \left(\sin k \cdot t\right)\right)\right)}} \]

   6. Simplified 78.3%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(k \cdot \left(k \cdot \left(\sin k \cdot t\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{k \cdot \left(\frac{k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)\right)}\\ \end{array} \]

Alternative 7: 68.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{1}{k \cdot \left(\frac{k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)}\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-65}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\cos k \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \frac{0.3333333333333333}{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k}\right)}^{2}}{{t}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -2.8e-29)
   (/ 1.0 (* k (* (/ k l) (/ (pow t 3.0) l))))
   (if (<= t 1.5e-65)
     (*
      2.0
      (*
       (* (/ l k) (/ l k))
       (* (cos k) (+ (/ 1.0 (* t (* k k))) (/ 0.3333333333333333 t)))))
     (/ (pow (/ l k) 2.0) (pow t 3.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -2.8e-29) {
		tmp = 1.0 / (k * ((k / l) * (pow(t, 3.0) / l)));
	} else if (t <= 1.5e-65) {
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) * ((1.0 / (t * (k * k))) + (0.3333333333333333 / t))));
	} else {
		tmp = pow((l / k), 2.0) / pow(t, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-2.8d-29)) then
        tmp = 1.0d0 / (k * ((k / l) * ((t ** 3.0d0) / l)))
    else if (t <= 1.5d-65) then
        tmp = 2.0d0 * (((l / k) * (l / k)) * (cos(k) * ((1.0d0 / (t * (k * k))) + (0.3333333333333333d0 / t))))
    else
        tmp = ((l / k) ** 2.0d0) / (t ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -2.8e-29) {
		tmp = 1.0 / (k * ((k / l) * (Math.pow(t, 3.0) / l)));
	} else if (t <= 1.5e-65) {
		tmp = 2.0 * (((l / k) * (l / k)) * (Math.cos(k) * ((1.0 / (t * (k * k))) + (0.3333333333333333 / t))));
	} else {
		tmp = Math.pow((l / k), 2.0) / Math.pow(t, 3.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -2.8e-29:
		tmp = 1.0 / (k * ((k / l) * (math.pow(t, 3.0) / l)))
	elif t <= 1.5e-65:
		tmp = 2.0 * (((l / k) * (l / k)) * (math.cos(k) * ((1.0 / (t * (k * k))) + (0.3333333333333333 / t))))
	else:
		tmp = math.pow((l / k), 2.0) / math.pow(t, 3.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -2.8e-29)
		tmp = Float64(1.0 / Float64(k * Float64(Float64(k / l) * Float64((t ^ 3.0) / l))));
	elseif (t <= 1.5e-65)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) * Float64(Float64(1.0 / Float64(t * Float64(k * k))) + Float64(0.3333333333333333 / t)))));
	else
		tmp = Float64((Float64(l / k) ^ 2.0) / (t ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -2.8e-29)
		tmp = 1.0 / (k * ((k / l) * ((t ^ 3.0) / l)));
	elseif (t <= 1.5e-65)
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) * ((1.0 / (t * (k * k))) + (0.3333333333333333 / t))));
	else
		tmp = ((l / k) ^ 2.0) / (t ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -2.8e-29], N[(1.0 / N[(k * N[(N[(k / l), $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.5e-65], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * N[(N[(1.0 / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.8000000000000002e-29

   1. Initial program 64.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 64.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. +-commutative 64.3%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 64.3%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

   4. Taylor expanded in k around 0 55.5%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-/l* 55.4%

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{{t}^{3}}}}} \]

      2. unpow2 55.4%

         \[\leadsto \frac{2}{2 \cdot \frac{\color{blue}{k \cdot k}}{\frac{{\ell}^{2}}{{t}^{3}}}} \]

      3. unpow2 55.4%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}}} \]

      4. associate-/l* 59.0%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}} \]

   6. Simplified 59.0%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{k \cdot k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 47.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{2 \cdot \frac{k \cdot k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}\right)\right)} \]

      2. expm1-udef 44.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{2 \cdot \frac{k \cdot k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}\right)} - 1} \]

      3. associate-/r* 44.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{2}{2}}{\frac{k \cdot k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}}\right)} - 1 \]

      4. metadata-eval 44.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\frac{k \cdot k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}\right)} - 1 \]

      5. associate-/r/ 44.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\frac{k \cdot k}{\ell} \cdot \frac{{t}^{3}}{\ell}}}\right)} - 1 \]

   8. Applied egg-rr 44.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\frac{k \cdot k}{\ell} \cdot \frac{{t}^{3}}{\ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 47.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{k \cdot k}{\ell} \cdot \frac{{t}^{3}}{\ell}}\right)\right)} \]

      2. expm1-log1p 57.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{k \cdot k}{\ell} \cdot \frac{{t}^{3}}{\ell}}} \]

      3. unpow2 57.5%

         \[\leadsto \frac{1}{\frac{\color{blue}{{k}^{2}}}{\ell} \cdot \frac{{t}^{3}}{\ell}} \]

      4. times-frac 55.5%

         \[\leadsto \frac{1}{\color{blue}{\frac{{k}^{2} \cdot {t}^{3}}{\ell \cdot \ell}}} \]

      5. unpow2 55.5%

         \[\leadsto \frac{1}{\frac{{k}^{2} \cdot {t}^{3}}{\color{blue}{{\ell}^{2}}}} \]

      6. associate-/l* 55.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{{t}^{3}}}}} \]

      7. unpow2 55.4%

         \[\leadsto \frac{1}{\frac{{k}^{2}}{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}}} \]

      8. associate-*r/ 59.0%

         \[\leadsto \frac{1}{\frac{{k}^{2}}{\color{blue}{\ell \cdot \frac{\ell}{{t}^{3}}}}} \]

      9. unpow2 59.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{k \cdot k}}{\ell \cdot \frac{\ell}{{t}^{3}}}} \]

      10. associate-*r/ 67.4%

          \[\leadsto \frac{1}{\color{blue}{k \cdot \frac{k}{\ell \cdot \frac{\ell}{{t}^{3}}}}} \]

      11. associate-*r/ 63.8%

          \[\leadsto \frac{1}{k \cdot \frac{k}{\color{blue}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]

      12. unpow2 63.8%

          \[\leadsto \frac{1}{k \cdot \frac{k}{\frac{\color{blue}{{\ell}^{2}}}{{t}^{3}}}} \]

      13. associate-/l* 65.8%

          \[\leadsto \frac{1}{k \cdot \color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}}} \]

      14. unpow2 65.8%

          \[\leadsto \frac{1}{k \cdot \frac{k \cdot {t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      15. times-frac 67.9%

          \[\leadsto \frac{1}{k \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)}} \]

      16. *-commutative 67.9%

          \[\leadsto \frac{1}{k \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right)}} \]

   10. Simplified 67.9%

       \[\leadsto \color{blue}{\frac{1}{k \cdot \left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right)}} \]

   if -2.8000000000000002e-29 < t < 1.49999999999999999e-65

   1. Initial program 36.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 36.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 36.1%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 36.0%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 36.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 36.0%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 36.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 36.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 36.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 36.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 36.0%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 36.0%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 66.7%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 66.7%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 66.6%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 66.6%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 66.6%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 87.4%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   6. Simplified 87.4%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
   7. Step-by-step derivation

      1. div-inv 87.3%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\cos k \cdot \frac{1}{{\sin k}^{2} \cdot t}\right)}\right) \]

   8. Applied egg-rr 87.3%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\cos k \cdot \frac{1}{{\sin k}^{2} \cdot t}\right)}\right) \]

   9. Taylor expanded in k around 0 68.4%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\cos k \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right)}\right)\right) \]
   10. Step-by-step derivation

       1. +-commutative 68.4%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\cos k \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{1}{t}\right)}\right)\right) \]

       2. unpow2 68.4%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\cos k \cdot \left(\frac{1}{\color{blue}{\left(k \cdot k\right)} \cdot t} + 0.3333333333333333 \cdot \frac{1}{t}\right)\right)\right) \]

       3. associate-*r/ 68.4%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\cos k \cdot \left(\frac{1}{\left(k \cdot k\right) \cdot t} + \color{blue}{\frac{0.3333333333333333 \cdot 1}{t}}\right)\right)\right) \]

       4. metadata-eval 68.4%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\cos k \cdot \left(\frac{1}{\left(k \cdot k\right) \cdot t} + \frac{\color{blue}{0.3333333333333333}}{t}\right)\right)\right) \]

   11. Simplified 68.4%

       \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\cos k \cdot \color{blue}{\left(\frac{1}{\left(k \cdot k\right) \cdot t} + \frac{0.3333333333333333}{t}\right)}\right)\right) \]

   if 1.49999999999999999e-65 < t

   1. Initial program 56.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 56.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 57.9%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 57.1%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 56.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 56.7%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 56.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 56.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 56.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 56.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 56.7%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 56.7%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around 0 45.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 45.6%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 45.6%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac 49.6%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. unpow2 49.6%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   6. Simplified 49.6%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]

   7. Taylor expanded in l around 0 45.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   8. Step-by-step derivation

      1. associate-/r* 45.6%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}} \]

      2. unpow2 45.6%

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{t}^{3}} \]

      3. unpow2 45.6%

         \[\leadsto \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{{t}^{3}} \]

      4. times-frac 62.5%

         \[\leadsto \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{{t}^{3}} \]

      5. unpow2 62.5%

         \[\leadsto \frac{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}{{t}^{3}} \]

   9. Simplified 62.5%

      \[\leadsto \color{blue}{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{{t}^{3}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{1}{k \cdot \left(\frac{k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)}\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-65}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\cos k \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \frac{0.3333333333333333}{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k}\right)}^{2}}{{t}^{3}}\\ \end{array} \]

Alternative 8: 69.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-33} \lor \neg \left(t \leq 4.4 \cdot 10^{-64}\right):\\ \;\;\;\;\frac{1}{k \cdot \left(\frac{k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\cos k \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \frac{0.3333333333333333}{t}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -1.15e-33) (not (<= t 4.4e-64)))
   (/ 1.0 (* k (* (/ k l) (/ (pow t 3.0) l))))
   (*
    2.0
    (*
     (* (/ l k) (/ l k))
     (* (cos k) (+ (/ 1.0 (* t (* k k))) (/ 0.3333333333333333 t)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.15e-33) || !(t <= 4.4e-64)) {
		tmp = 1.0 / (k * ((k / l) * (pow(t, 3.0) / l)));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) * ((1.0 / (t * (k * k))) + (0.3333333333333333 / t))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-1.15d-33)) .or. (.not. (t <= 4.4d-64))) then
        tmp = 1.0d0 / (k * ((k / l) * ((t ** 3.0d0) / l)))
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) * (cos(k) * ((1.0d0 / (t * (k * k))) + (0.3333333333333333d0 / t))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.15e-33) || !(t <= 4.4e-64)) {
		tmp = 1.0 / (k * ((k / l) * (Math.pow(t, 3.0) / l)));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (Math.cos(k) * ((1.0 / (t * (k * k))) + (0.3333333333333333 / t))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -1.15e-33) or not (t <= 4.4e-64):
		tmp = 1.0 / (k * ((k / l) * (math.pow(t, 3.0) / l)))
	else:
		tmp = 2.0 * (((l / k) * (l / k)) * (math.cos(k) * ((1.0 / (t * (k * k))) + (0.3333333333333333 / t))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -1.15e-33) || !(t <= 4.4e-64))
		tmp = Float64(1.0 / Float64(k * Float64(Float64(k / l) * Float64((t ^ 3.0) / l))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) * Float64(Float64(1.0 / Float64(t * Float64(k * k))) + Float64(0.3333333333333333 / t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -1.15e-33) || ~((t <= 4.4e-64)))
		tmp = 1.0 / (k * ((k / l) * ((t ^ 3.0) / l)));
	else
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) * ((1.0 / (t * (k * k))) + (0.3333333333333333 / t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -1.15e-33], N[Not[LessEqual[t, 4.4e-64]], $MachinePrecision]], N[(1.0 / N[(k * N[(N[(k / l), $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * N[(N[(1.0 / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.14999999999999993e-33 or 4.3999999999999999e-64 < t

   1. Initial program 60.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 60.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. +-commutative 60.0%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 60.0%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

   4. Taylor expanded in k around 0 49.9%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-/l* 49.2%

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{{t}^{3}}}}} \]

      2. unpow2 49.2%

         \[\leadsto \frac{2}{2 \cdot \frac{\color{blue}{k \cdot k}}{\frac{{\ell}^{2}}{{t}^{3}}}} \]

      3. unpow2 49.2%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}}} \]

      4. associate-/l* 53.9%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}} \]

   6. Simplified 53.9%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{k \cdot k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 48.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{2 \cdot \frac{k \cdot k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}\right)\right)} \]

      2. expm1-udef 46.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{2 \cdot \frac{k \cdot k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}\right)} - 1} \]

      3. associate-/r* 46.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{2}{2}}{\frac{k \cdot k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}}\right)} - 1 \]

      4. metadata-eval 46.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\frac{k \cdot k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}\right)} - 1 \]

      5. associate-/r/ 46.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\frac{k \cdot k}{\ell} \cdot \frac{{t}^{3}}{\ell}}}\right)} - 1 \]

   8. Applied egg-rr 46.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\frac{k \cdot k}{\ell} \cdot \frac{{t}^{3}}{\ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 48.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{k \cdot k}{\ell} \cdot \frac{{t}^{3}}{\ell}}\right)\right)} \]

      2. expm1-log1p 53.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{k \cdot k}{\ell} \cdot \frac{{t}^{3}}{\ell}}} \]

      3. unpow2 53.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{{k}^{2}}}{\ell} \cdot \frac{{t}^{3}}{\ell}} \]

      4. times-frac 49.9%

         \[\leadsto \frac{1}{\color{blue}{\frac{{k}^{2} \cdot {t}^{3}}{\ell \cdot \ell}}} \]

      5. unpow2 49.9%

         \[\leadsto \frac{1}{\frac{{k}^{2} \cdot {t}^{3}}{\color{blue}{{\ell}^{2}}}} \]

      6. associate-/l* 49.2%

         \[\leadsto \frac{1}{\color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{{t}^{3}}}}} \]

      7. unpow2 49.2%

         \[\leadsto \frac{1}{\frac{{k}^{2}}{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}}} \]

      8. associate-*r/ 53.9%

         \[\leadsto \frac{1}{\frac{{k}^{2}}{\color{blue}{\ell \cdot \frac{\ell}{{t}^{3}}}}} \]

      9. unpow2 53.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{k \cdot k}}{\ell \cdot \frac{\ell}{{t}^{3}}}} \]

      10. associate-*r/ 63.3%

          \[\leadsto \frac{1}{\color{blue}{k \cdot \frac{k}{\ell \cdot \frac{\ell}{{t}^{3}}}}} \]

      11. associate-*r/ 58.6%

          \[\leadsto \frac{1}{k \cdot \frac{k}{\color{blue}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]

      12. unpow2 58.6%

          \[\leadsto \frac{1}{k \cdot \frac{k}{\frac{\color{blue}{{\ell}^{2}}}{{t}^{3}}}} \]

      13. associate-/l* 60.1%

          \[\leadsto \frac{1}{k \cdot \color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}}} \]

      14. unpow2 60.1%

          \[\leadsto \frac{1}{k \cdot \frac{k \cdot {t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      15. times-frac 64.7%

          \[\leadsto \frac{1}{k \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)}} \]

      16. *-commutative 64.7%

          \[\leadsto \frac{1}{k \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right)}} \]

   10. Simplified 64.7%

       \[\leadsto \color{blue}{\frac{1}{k \cdot \left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right)}} \]

   if -1.14999999999999993e-33 < t < 4.3999999999999999e-64

   1. Initial program 36.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 36.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 36.1%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 36.0%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 36.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 36.0%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 36.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 36.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 36.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 36.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 36.0%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 36.0%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 66.7%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 66.7%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 66.6%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 66.6%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 66.6%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 87.4%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   6. Simplified 87.4%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
   7. Step-by-step derivation

      1. div-inv 87.3%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\cos k \cdot \frac{1}{{\sin k}^{2} \cdot t}\right)}\right) \]

   8. Applied egg-rr 87.3%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\cos k \cdot \frac{1}{{\sin k}^{2} \cdot t}\right)}\right) \]

   9. Taylor expanded in k around 0 68.4%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\cos k \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right)}\right)\right) \]
   10. Step-by-step derivation

       1. +-commutative 68.4%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\cos k \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{1}{t}\right)}\right)\right) \]

       2. unpow2 68.4%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\cos k \cdot \left(\frac{1}{\color{blue}{\left(k \cdot k\right)} \cdot t} + 0.3333333333333333 \cdot \frac{1}{t}\right)\right)\right) \]

       3. associate-*r/ 68.4%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\cos k \cdot \left(\frac{1}{\left(k \cdot k\right) \cdot t} + \color{blue}{\frac{0.3333333333333333 \cdot 1}{t}}\right)\right)\right) \]

       4. metadata-eval 68.4%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\cos k \cdot \left(\frac{1}{\left(k \cdot k\right) \cdot t} + \frac{\color{blue}{0.3333333333333333}}{t}\right)\right)\right) \]

   11. Simplified 68.4%

       \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\cos k \cdot \color{blue}{\left(\frac{1}{\left(k \cdot k\right) \cdot t} + \frac{0.3333333333333333}{t}\right)}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-33} \lor \neg \left(t \leq 4.4 \cdot 10^{-64}\right):\\ \;\;\;\;\frac{1}{k \cdot \left(\frac{k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\cos k \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \frac{0.3333333333333333}{t}\right)\right)\right)\\ \end{array} \]

Alternative 9: 62.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_2 := \left(\ell \cdot {t}^{-3}\right) \cdot \frac{\ell}{k \cdot k}\\ \mathbf{if}\;t \leq -1.1 \cdot 10^{-48}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-67}:\\ \;\;\;\;2 \cdot \left(\frac{t_1}{t} \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right)\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{+107}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{1}{k \cdot \left(t \cdot k\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ l k) (/ l k))) (t_2 (* (* l (pow t -3.0)) (/ l (* k k)))))
   (if (<= t -1.1e-48)
     t_2
     (if (<= t 9e-67)
       (* 2.0 (* (/ t_1 t) (+ (/ 1.0 (* k k)) -0.16666666666666666)))
       (if (<= t 2.35e+107) t_2 (* 2.0 (* t_1 (/ 1.0 (* k (* t k))))))))))

C

double code(double t, double l, double k) {
	double t_1 = (l / k) * (l / k);
	double t_2 = (l * pow(t, -3.0)) * (l / (k * k));
	double tmp;
	if (t <= -1.1e-48) {
		tmp = t_2;
	} else if (t <= 9e-67) {
		tmp = 2.0 * ((t_1 / t) * ((1.0 / (k * k)) + -0.16666666666666666));
	} else if (t <= 2.35e+107) {
		tmp = t_2;
	} else {
		tmp = 2.0 * (t_1 * (1.0 / (k * (t * k))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (l / k) * (l / k)
    t_2 = (l * (t ** (-3.0d0))) * (l / (k * k))
    if (t <= (-1.1d-48)) then
        tmp = t_2
    else if (t <= 9d-67) then
        tmp = 2.0d0 * ((t_1 / t) * ((1.0d0 / (k * k)) + (-0.16666666666666666d0)))
    else if (t <= 2.35d+107) then
        tmp = t_2
    else
        tmp = 2.0d0 * (t_1 * (1.0d0 / (k * (t * k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = (l / k) * (l / k);
	double t_2 = (l * Math.pow(t, -3.0)) * (l / (k * k));
	double tmp;
	if (t <= -1.1e-48) {
		tmp = t_2;
	} else if (t <= 9e-67) {
		tmp = 2.0 * ((t_1 / t) * ((1.0 / (k * k)) + -0.16666666666666666));
	} else if (t <= 2.35e+107) {
		tmp = t_2;
	} else {
		tmp = 2.0 * (t_1 * (1.0 / (k * (t * k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (l / k) * (l / k)
	t_2 = (l * math.pow(t, -3.0)) * (l / (k * k))
	tmp = 0
	if t <= -1.1e-48:
		tmp = t_2
	elif t <= 9e-67:
		tmp = 2.0 * ((t_1 / t) * ((1.0 / (k * k)) + -0.16666666666666666))
	elif t <= 2.35e+107:
		tmp = t_2
	else:
		tmp = 2.0 * (t_1 * (1.0 / (k * (t * k))))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(l / k) * Float64(l / k))
	t_2 = Float64(Float64(l * (t ^ -3.0)) * Float64(l / Float64(k * k)))
	tmp = 0.0
	if (t <= -1.1e-48)
		tmp = t_2;
	elseif (t <= 9e-67)
		tmp = Float64(2.0 * Float64(Float64(t_1 / t) * Float64(Float64(1.0 / Float64(k * k)) + -0.16666666666666666)));
	elseif (t <= 2.35e+107)
		tmp = t_2;
	else
		tmp = Float64(2.0 * Float64(t_1 * Float64(1.0 / Float64(k * Float64(t * k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (l / k) * (l / k);
	t_2 = (l * (t ^ -3.0)) * (l / (k * k));
	tmp = 0.0;
	if (t <= -1.1e-48)
		tmp = t_2;
	elseif (t <= 9e-67)
		tmp = 2.0 * ((t_1 / t) * ((1.0 / (k * k)) + -0.16666666666666666));
	elseif (t <= 2.35e+107)
		tmp = t_2;
	else
		tmp = 2.0 * (t_1 * (1.0 / (k * (t * k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l * N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.1e-48], t$95$2, If[LessEqual[t, 9e-67], N[(2.0 * N[(N[(t$95$1 / t), $MachinePrecision] * N[(N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.35e+107], t$95$2, N[(2.0 * N[(t$95$1 * N[(1.0 / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.10000000000000006e-48 or 9.00000000000000031e-67 < t < 2.35e107

   1. Initial program 67.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 67.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 70.6%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 68.7%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 68.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 68.4%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 68.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 68.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 68.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 68.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 68.4%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 68.4%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around 0 58.0%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 58.0%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 58.0%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac 61.2%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. unpow2 61.2%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   6. Simplified 61.2%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 56.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell}{{t}^{3}}\right)\right)} \cdot \frac{\ell}{k \cdot k} \]

      2. expm1-udef 48.6%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\ell}{{t}^{3}}\right)} - 1\right)} \cdot \frac{\ell}{k \cdot k} \]

      3. div-inv 48.6%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\ell \cdot \frac{1}{{t}^{3}}}\right)} - 1\right) \cdot \frac{\ell}{k \cdot k} \]

      4. pow-flip 48.6%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\ell \cdot \color{blue}{{t}^{\left(-3\right)}}\right)} - 1\right) \cdot \frac{\ell}{k \cdot k} \]

      5. metadata-eval 48.6%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\ell \cdot {t}^{\color{blue}{-3}}\right)} - 1\right) \cdot \frac{\ell}{k \cdot k} \]

   8. Applied egg-rr 48.6%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\ell \cdot {t}^{-3}\right)} - 1\right)} \cdot \frac{\ell}{k \cdot k} \]
   9. Step-by-step derivation

      1. expm1-def 56.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot {t}^{-3}\right)\right)} \cdot \frac{\ell}{k \cdot k} \]

      2. expm1-log1p 61.2%

         \[\leadsto \color{blue}{\left(\ell \cdot {t}^{-3}\right)} \cdot \frac{\ell}{k \cdot k} \]

   10. Simplified 61.2%

       \[\leadsto \color{blue}{\left(\ell \cdot {t}^{-3}\right)} \cdot \frac{\ell}{k \cdot k} \]

   if -1.10000000000000006e-48 < t < 9.00000000000000031e-67

   1. Initial program 34.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 34.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 34.8%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 34.8%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 34.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 34.9%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 34.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 34.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 34.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 34.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 34.9%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 34.9%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 67.0%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 67.0%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 66.8%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 66.8%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 66.8%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 87.7%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   6. Simplified 87.7%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

   7. Taylor expanded in l around 0 67.0%

      \[\leadsto 2 \cdot \color{blue}{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* 67.1%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{{\sin k}^{2} \cdot t}} \]

      2. associate-*r/ 67.1%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot \frac{{\ell}^{2}}{{k}^{2}}}}{{\sin k}^{2} \cdot t} \]

      3. unpow2 67.1%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{\sin k}^{2} \cdot t} \]

      4. unpow2 67.1%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{{\sin k}^{2} \cdot t} \]

      5. times-frac 88.0%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}}{{\sin k}^{2} \cdot t} \]

      6. unpow2 88.0%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}{{\sin k}^{2} \cdot t} \]

      7. *-commutative 88.0%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}}{{\sin k}^{2} \cdot t} \]

      8. *-commutative 88.0%

         \[\leadsto 2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}{\color{blue}{t \cdot {\sin k}^{2}}} \]

      9. times-frac 88.7%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   9. Simplified 88.7%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   10. Taylor expanded in k around 0 64.6%

       \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \color{blue}{\left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)}\right) \]
   11. Step-by-step derivation

       1. sub-neg 64.6%

          \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \color{blue}{\left(\frac{1}{{k}^{2}} + \left(-0.16666666666666666\right)\right)}\right) \]

       2. unpow2 64.6%

          \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \left(\frac{1}{\color{blue}{k \cdot k}} + \left(-0.16666666666666666\right)\right)\right) \]

       3. metadata-eval 64.6%

          \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \left(\frac{1}{k \cdot k} + \color{blue}{-0.16666666666666666}\right)\right) \]

   12. Simplified 64.6%

       \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \color{blue}{\left(\frac{1}{k \cdot k} + -0.16666666666666666\right)}\right) \]
   13. Step-by-step derivation

       1. pow2 64.6%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right) \]

   14. Applied egg-rr 64.6%

       \[\leadsto 2 \cdot \left(\frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right) \]

   if 2.35e107 < t

   1. Initial program 41.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 41.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 41.6%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 41.6%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 41.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 41.6%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 41.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 41.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 41.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 41.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 41.6%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 41.6%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 43.6%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 43.6%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 41.6%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 41.6%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 41.6%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 44.0%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   6. Simplified 44.0%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

   7. Taylor expanded in k around 0 45.9%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{1}{{k}^{2} \cdot t}}\right) \]
   8. Step-by-step derivation

      1. unpow2 45.9%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right) \]

      2. associate-*l* 45.9%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{\color{blue}{k \cdot \left(k \cdot t\right)}}\right) \]

   9. Simplified 45.9%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{1}{k \cdot \left(k \cdot t\right)}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{-48}:\\ \;\;\;\;\left(\ell \cdot {t}^{-3}\right) \cdot \frac{\ell}{k \cdot k}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-67}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right)\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{+107}:\\ \;\;\;\;\left(\ell \cdot {t}^{-3}\right) \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{k \cdot \left(t \cdot k\right)}\right)\\ \end{array} \]

Alternative 10: 62.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_2 := \frac{\ell}{k \cdot k}\\ \mathbf{if}\;t \leq -4.6 \cdot 10^{-50}:\\ \;\;\;\;\left(\ell \cdot {t}^{-3}\right) \cdot t_2\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-64}:\\ \;\;\;\;2 \cdot \left(\frac{t_1}{t} \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right)\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot t_2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{1}{k \cdot \left(t \cdot k\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ l k) (/ l k))) (t_2 (/ l (* k k))))
   (if (<= t -4.6e-50)
     (* (* l (pow t -3.0)) t_2)
     (if (<= t 1.6e-64)
       (* 2.0 (* (/ t_1 t) (+ (/ 1.0 (* k k)) -0.16666666666666666)))
       (if (<= t 6.5e+102)
         (* (/ l (pow t 3.0)) t_2)
         (* 2.0 (* t_1 (/ 1.0 (* k (* t k))))))))))

C

double code(double t, double l, double k) {
	double t_1 = (l / k) * (l / k);
	double t_2 = l / (k * k);
	double tmp;
	if (t <= -4.6e-50) {
		tmp = (l * pow(t, -3.0)) * t_2;
	} else if (t <= 1.6e-64) {
		tmp = 2.0 * ((t_1 / t) * ((1.0 / (k * k)) + -0.16666666666666666));
	} else if (t <= 6.5e+102) {
		tmp = (l / pow(t, 3.0)) * t_2;
	} else {
		tmp = 2.0 * (t_1 * (1.0 / (k * (t * k))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (l / k) * (l / k)
    t_2 = l / (k * k)
    if (t <= (-4.6d-50)) then
        tmp = (l * (t ** (-3.0d0))) * t_2
    else if (t <= 1.6d-64) then
        tmp = 2.0d0 * ((t_1 / t) * ((1.0d0 / (k * k)) + (-0.16666666666666666d0)))
    else if (t <= 6.5d+102) then
        tmp = (l / (t ** 3.0d0)) * t_2
    else
        tmp = 2.0d0 * (t_1 * (1.0d0 / (k * (t * k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = (l / k) * (l / k);
	double t_2 = l / (k * k);
	double tmp;
	if (t <= -4.6e-50) {
		tmp = (l * Math.pow(t, -3.0)) * t_2;
	} else if (t <= 1.6e-64) {
		tmp = 2.0 * ((t_1 / t) * ((1.0 / (k * k)) + -0.16666666666666666));
	} else if (t <= 6.5e+102) {
		tmp = (l / Math.pow(t, 3.0)) * t_2;
	} else {
		tmp = 2.0 * (t_1 * (1.0 / (k * (t * k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (l / k) * (l / k)
	t_2 = l / (k * k)
	tmp = 0
	if t <= -4.6e-50:
		tmp = (l * math.pow(t, -3.0)) * t_2
	elif t <= 1.6e-64:
		tmp = 2.0 * ((t_1 / t) * ((1.0 / (k * k)) + -0.16666666666666666))
	elif t <= 6.5e+102:
		tmp = (l / math.pow(t, 3.0)) * t_2
	else:
		tmp = 2.0 * (t_1 * (1.0 / (k * (t * k))))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(l / k) * Float64(l / k))
	t_2 = Float64(l / Float64(k * k))
	tmp = 0.0
	if (t <= -4.6e-50)
		tmp = Float64(Float64(l * (t ^ -3.0)) * t_2);
	elseif (t <= 1.6e-64)
		tmp = Float64(2.0 * Float64(Float64(t_1 / t) * Float64(Float64(1.0 / Float64(k * k)) + -0.16666666666666666)));
	elseif (t <= 6.5e+102)
		tmp = Float64(Float64(l / (t ^ 3.0)) * t_2);
	else
		tmp = Float64(2.0 * Float64(t_1 * Float64(1.0 / Float64(k * Float64(t * k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (l / k) * (l / k);
	t_2 = l / (k * k);
	tmp = 0.0;
	if (t <= -4.6e-50)
		tmp = (l * (t ^ -3.0)) * t_2;
	elseif (t <= 1.6e-64)
		tmp = 2.0 * ((t_1 / t) * ((1.0 / (k * k)) + -0.16666666666666666));
	elseif (t <= 6.5e+102)
		tmp = (l / (t ^ 3.0)) * t_2;
	else
		tmp = 2.0 * (t_1 * (1.0 / (k * (t * k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4.6e-50], N[(N[(l * N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t, 1.6e-64], N[(2.0 * N[(N[(t$95$1 / t), $MachinePrecision] * N[(N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.5e+102], N[(N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(2.0 * N[(t$95$1 * N[(1.0 / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -4.60000000000000039e-50

   1. Initial program 64.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 64.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 67.0%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 65.0%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 65.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 65.0%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 65.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 65.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 65.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 65.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 65.0%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 65.0%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around 0 55.9%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 55.9%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 55.9%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac 59.1%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. unpow2 59.1%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   6. Simplified 59.1%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 54.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell}{{t}^{3}}\right)\right)} \cdot \frac{\ell}{k \cdot k} \]

      2. expm1-udef 45.9%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\ell}{{t}^{3}}\right)} - 1\right)} \cdot \frac{\ell}{k \cdot k} \]

      3. div-inv 45.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\ell \cdot \frac{1}{{t}^{3}}}\right)} - 1\right) \cdot \frac{\ell}{k \cdot k} \]

      4. pow-flip 45.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\ell \cdot \color{blue}{{t}^{\left(-3\right)}}\right)} - 1\right) \cdot \frac{\ell}{k \cdot k} \]

      5. metadata-eval 45.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\ell \cdot {t}^{\color{blue}{-3}}\right)} - 1\right) \cdot \frac{\ell}{k \cdot k} \]

   8. Applied egg-rr 45.9%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\ell \cdot {t}^{-3}\right)} - 1\right)} \cdot \frac{\ell}{k \cdot k} \]
   9. Step-by-step derivation

      1. expm1-def 54.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot {t}^{-3}\right)\right)} \cdot \frac{\ell}{k \cdot k} \]

      2. expm1-log1p 59.2%

         \[\leadsto \color{blue}{\left(\ell \cdot {t}^{-3}\right)} \cdot \frac{\ell}{k \cdot k} \]

   10. Simplified 59.2%

       \[\leadsto \color{blue}{\left(\ell \cdot {t}^{-3}\right)} \cdot \frac{\ell}{k \cdot k} \]

   if -4.60000000000000039e-50 < t < 1.59999999999999988e-64

   1. Initial program 34.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 34.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 34.8%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 34.8%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 34.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 34.9%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 34.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 34.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 34.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 34.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 34.9%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 34.9%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 67.0%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 67.0%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 66.8%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 66.8%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 66.8%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 87.7%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   6. Simplified 87.7%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

   7. Taylor expanded in l around 0 67.0%

      \[\leadsto 2 \cdot \color{blue}{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* 67.1%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{{\sin k}^{2} \cdot t}} \]

      2. associate-*r/ 67.1%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot \frac{{\ell}^{2}}{{k}^{2}}}}{{\sin k}^{2} \cdot t} \]

      3. unpow2 67.1%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{\sin k}^{2} \cdot t} \]

      4. unpow2 67.1%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{{\sin k}^{2} \cdot t} \]

      5. times-frac 88.0%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}}{{\sin k}^{2} \cdot t} \]

      6. unpow2 88.0%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}{{\sin k}^{2} \cdot t} \]

      7. *-commutative 88.0%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}}{{\sin k}^{2} \cdot t} \]

      8. *-commutative 88.0%

         \[\leadsto 2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}{\color{blue}{t \cdot {\sin k}^{2}}} \]

      9. times-frac 88.7%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   9. Simplified 88.7%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   10. Taylor expanded in k around 0 64.6%

       \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \color{blue}{\left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)}\right) \]
   11. Step-by-step derivation

       1. sub-neg 64.6%

          \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \color{blue}{\left(\frac{1}{{k}^{2}} + \left(-0.16666666666666666\right)\right)}\right) \]

       2. unpow2 64.6%

          \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \left(\frac{1}{\color{blue}{k \cdot k}} + \left(-0.16666666666666666\right)\right)\right) \]

       3. metadata-eval 64.6%

          \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \left(\frac{1}{k \cdot k} + \color{blue}{-0.16666666666666666}\right)\right) \]

   12. Simplified 64.6%

       \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \color{blue}{\left(\frac{1}{k \cdot k} + -0.16666666666666666\right)}\right) \]
   13. Step-by-step derivation

       1. pow2 64.6%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right) \]

   14. Applied egg-rr 64.6%

       \[\leadsto 2 \cdot \left(\frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right) \]

   if 1.59999999999999988e-64 < t < 6.5000000000000004e102

   1. Initial program 74.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 74.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 77.3%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 75.5%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 74.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 74.6%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 74.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 74.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 74.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 74.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 74.5%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 74.5%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around 0 61.9%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 61.9%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 61.9%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac 64.9%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. unpow2 64.9%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   6. Simplified 64.9%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]

   if 6.5000000000000004e102 < t

   1. Initial program 41.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 41.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 41.6%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 41.6%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 41.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 41.6%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 41.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 41.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 41.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 41.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 41.6%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 41.6%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 43.6%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 43.6%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 41.6%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 41.6%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 41.6%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 44.0%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   6. Simplified 44.0%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

   7. Taylor expanded in k around 0 45.9%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{1}{{k}^{2} \cdot t}}\right) \]
   8. Step-by-step derivation

      1. unpow2 45.9%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right) \]

      2. associate-*l* 45.9%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{\color{blue}{k \cdot \left(k \cdot t\right)}}\right) \]

   9. Simplified 45.9%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{1}{k \cdot \left(k \cdot t\right)}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.6 \cdot 10^{-50}:\\ \;\;\;\;\left(\ell \cdot {t}^{-3}\right) \cdot \frac{\ell}{k \cdot k}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-64}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right)\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{k \cdot \left(t \cdot k\right)}\right)\\ \end{array} \]

Alternative 11: 67.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{-45} \lor \neg \left(t \leq 2.25 \cdot 10^{-98}\right):\\ \;\;\;\;\frac{1}{k \cdot \left(\frac{k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -3.6e-45) (not (<= t 2.25e-98)))
   (/ 1.0 (* k (* (/ k l) (/ (pow t 3.0) l))))
   (*
    2.0
    (* (/ (* (/ l k) (/ l k)) t) (+ (/ 1.0 (* k k)) -0.16666666666666666)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -3.6e-45) || !(t <= 2.25e-98)) {
		tmp = 1.0 / (k * ((k / l) * (pow(t, 3.0) / l)));
	} else {
		tmp = 2.0 * ((((l / k) * (l / k)) / t) * ((1.0 / (k * k)) + -0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-3.6d-45)) .or. (.not. (t <= 2.25d-98))) then
        tmp = 1.0d0 / (k * ((k / l) * ((t ** 3.0d0) / l)))
    else
        tmp = 2.0d0 * ((((l / k) * (l / k)) / t) * ((1.0d0 / (k * k)) + (-0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -3.6e-45) || !(t <= 2.25e-98)) {
		tmp = 1.0 / (k * ((k / l) * (Math.pow(t, 3.0) / l)));
	} else {
		tmp = 2.0 * ((((l / k) * (l / k)) / t) * ((1.0 / (k * k)) + -0.16666666666666666));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -3.6e-45) or not (t <= 2.25e-98):
		tmp = 1.0 / (k * ((k / l) * (math.pow(t, 3.0) / l)))
	else:
		tmp = 2.0 * ((((l / k) * (l / k)) / t) * ((1.0 / (k * k)) + -0.16666666666666666))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -3.6e-45) || !(t <= 2.25e-98))
		tmp = Float64(1.0 / Float64(k * Float64(Float64(k / l) * Float64((t ^ 3.0) / l))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(l / k) * Float64(l / k)) / t) * Float64(Float64(1.0 / Float64(k * k)) + -0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -3.6e-45) || ~((t <= 2.25e-98)))
		tmp = 1.0 / (k * ((k / l) * ((t ^ 3.0) / l)));
	else
		tmp = 2.0 * ((((l / k) * (l / k)) / t) * ((1.0 / (k * k)) + -0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -3.6e-45], N[Not[LessEqual[t, 2.25e-98]], $MachinePrecision]], N[(1.0 / N[(k * N[(N[(k / l), $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.60000000000000001e-45 or 2.24999999999999998e-98 < t

   1. Initial program 60.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 60.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. +-commutative 60.8%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 60.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

   4. Taylor expanded in k around 0 51.2%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-/l* 50.6%

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{{t}^{3}}}}} \]

      2. unpow2 50.6%

         \[\leadsto \frac{2}{2 \cdot \frac{\color{blue}{k \cdot k}}{\frac{{\ell}^{2}}{{t}^{3}}}} \]

      3. unpow2 50.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}}} \]

      4. associate-/l* 55.0%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}} \]

   6. Simplified 55.0%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{k \cdot k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 48.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{2 \cdot \frac{k \cdot k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}\right)\right)} \]

      2. expm1-udef 47.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{2 \cdot \frac{k \cdot k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}\right)} - 1} \]

      3. associate-/r* 47.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{2}{2}}{\frac{k \cdot k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}}\right)} - 1 \]

      4. metadata-eval 47.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\frac{k \cdot k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}\right)} - 1 \]

      5. associate-/r/ 46.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\frac{k \cdot k}{\ell} \cdot \frac{{t}^{3}}{\ell}}}\right)} - 1 \]

   8. Applied egg-rr 46.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\frac{k \cdot k}{\ell} \cdot \frac{{t}^{3}}{\ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 48.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{k \cdot k}{\ell} \cdot \frac{{t}^{3}}{\ell}}\right)\right)} \]

      2. expm1-log1p 54.1%

         \[\leadsto \color{blue}{\frac{1}{\frac{k \cdot k}{\ell} \cdot \frac{{t}^{3}}{\ell}}} \]

      3. unpow2 54.1%

         \[\leadsto \frac{1}{\frac{\color{blue}{{k}^{2}}}{\ell} \cdot \frac{{t}^{3}}{\ell}} \]

      4. times-frac 51.2%

         \[\leadsto \frac{1}{\color{blue}{\frac{{k}^{2} \cdot {t}^{3}}{\ell \cdot \ell}}} \]

      5. unpow2 51.2%

         \[\leadsto \frac{1}{\frac{{k}^{2} \cdot {t}^{3}}{\color{blue}{{\ell}^{2}}}} \]

      6. associate-/l* 50.6%

         \[\leadsto \frac{1}{\color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{{t}^{3}}}}} \]

      7. unpow2 50.6%

         \[\leadsto \frac{1}{\frac{{k}^{2}}{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}}} \]

      8. associate-*r/ 55.0%

         \[\leadsto \frac{1}{\frac{{k}^{2}}{\color{blue}{\ell \cdot \frac{\ell}{{t}^{3}}}}} \]

      9. unpow2 55.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{k \cdot k}}{\ell \cdot \frac{\ell}{{t}^{3}}}} \]

      10. associate-*r/ 63.9%

          \[\leadsto \frac{1}{\color{blue}{k \cdot \frac{k}{\ell \cdot \frac{\ell}{{t}^{3}}}}} \]

      11. associate-*r/ 59.5%

          \[\leadsto \frac{1}{k \cdot \frac{k}{\color{blue}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]

      12. unpow2 59.5%

          \[\leadsto \frac{1}{k \cdot \frac{k}{\frac{\color{blue}{{\ell}^{2}}}{{t}^{3}}}} \]

      13. associate-/l* 60.9%

          \[\leadsto \frac{1}{k \cdot \color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}}} \]

      14. unpow2 60.9%

          \[\leadsto \frac{1}{k \cdot \frac{k \cdot {t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      15. times-frac 65.3%

          \[\leadsto \frac{1}{k \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)}} \]

      16. *-commutative 65.3%

          \[\leadsto \frac{1}{k \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right)}} \]

   10. Simplified 65.3%

       \[\leadsto \color{blue}{\frac{1}{k \cdot \left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right)}} \]

   if -3.60000000000000001e-45 < t < 2.24999999999999998e-98

   1. Initial program 33.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 33.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 33.0%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 32.9%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 32.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 32.9%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 32.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 32.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 32.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 32.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 32.9%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 32.9%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 66.0%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 66.0%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 65.9%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 65.9%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 65.9%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 86.4%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   6. Simplified 86.4%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

   7. Taylor expanded in l around 0 66.0%

      \[\leadsto 2 \cdot \color{blue}{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* 66.2%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{{\sin k}^{2} \cdot t}} \]

      2. associate-*r/ 66.2%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot \frac{{\ell}^{2}}{{k}^{2}}}}{{\sin k}^{2} \cdot t} \]

      3. unpow2 66.2%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{\sin k}^{2} \cdot t} \]

      4. unpow2 66.2%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{{\sin k}^{2} \cdot t} \]

      5. times-frac 86.7%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}}{{\sin k}^{2} \cdot t} \]

      6. unpow2 86.7%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}{{\sin k}^{2} \cdot t} \]

      7. *-commutative 86.7%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}}{{\sin k}^{2} \cdot t} \]

      8. *-commutative 86.7%

         \[\leadsto 2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}{\color{blue}{t \cdot {\sin k}^{2}}} \]

      9. times-frac 87.4%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   9. Simplified 87.4%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   10. Taylor expanded in k around 0 63.6%

       \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \color{blue}{\left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)}\right) \]
   11. Step-by-step derivation

       1. sub-neg 63.6%

          \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \color{blue}{\left(\frac{1}{{k}^{2}} + \left(-0.16666666666666666\right)\right)}\right) \]

       2. unpow2 63.6%

          \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \left(\frac{1}{\color{blue}{k \cdot k}} + \left(-0.16666666666666666\right)\right)\right) \]

       3. metadata-eval 63.6%

          \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \left(\frac{1}{k \cdot k} + \color{blue}{-0.16666666666666666}\right)\right) \]

   12. Simplified 63.6%

       \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \color{blue}{\left(\frac{1}{k \cdot k} + -0.16666666666666666\right)}\right) \]
   13. Step-by-step derivation

       1. pow2 63.6%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right) \]

   14. Applied egg-rr 63.6%

       \[\leadsto 2 \cdot \left(\frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{-45} \lor \neg \left(t \leq 2.25 \cdot 10^{-98}\right):\\ \;\;\;\;\frac{1}{k \cdot \left(\frac{k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right)\\ \end{array} \]

Alternative 12: 65.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.4 \cdot 10^{-45} \lor \neg \left(t \leq 1.9 \cdot 10^{-62}\right):\\ \;\;\;\;\frac{\ell \cdot \ell}{k \cdot \left({t}^{3} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -5.4e-45) (not (<= t 1.9e-62)))
   (/ (* l l) (* k (* (pow t 3.0) k)))
   (*
    2.0
    (* (/ (* (/ l k) (/ l k)) t) (+ (/ 1.0 (* k k)) -0.16666666666666666)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -5.4e-45) || !(t <= 1.9e-62)) {
		tmp = (l * l) / (k * (pow(t, 3.0) * k));
	} else {
		tmp = 2.0 * ((((l / k) * (l / k)) / t) * ((1.0 / (k * k)) + -0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-5.4d-45)) .or. (.not. (t <= 1.9d-62))) then
        tmp = (l * l) / (k * ((t ** 3.0d0) * k))
    else
        tmp = 2.0d0 * ((((l / k) * (l / k)) / t) * ((1.0d0 / (k * k)) + (-0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -5.4e-45) || !(t <= 1.9e-62)) {
		tmp = (l * l) / (k * (Math.pow(t, 3.0) * k));
	} else {
		tmp = 2.0 * ((((l / k) * (l / k)) / t) * ((1.0 / (k * k)) + -0.16666666666666666));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -5.4e-45) or not (t <= 1.9e-62):
		tmp = (l * l) / (k * (math.pow(t, 3.0) * k))
	else:
		tmp = 2.0 * ((((l / k) * (l / k)) / t) * ((1.0 / (k * k)) + -0.16666666666666666))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -5.4e-45) || !(t <= 1.9e-62))
		tmp = Float64(Float64(l * l) / Float64(k * Float64((t ^ 3.0) * k)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(l / k) * Float64(l / k)) / t) * Float64(Float64(1.0 / Float64(k * k)) + -0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -5.4e-45) || ~((t <= 1.9e-62)))
		tmp = (l * l) / (k * ((t ^ 3.0) * k));
	else
		tmp = 2.0 * ((((l / k) * (l / k)) / t) * ((1.0 / (k * k)) + -0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -5.4e-45], N[Not[LessEqual[t, 1.9e-62]], $MachinePrecision]], N[(N[(l * l), $MachinePrecision] / N[(k * N[(N[Power[t, 3.0], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.3999999999999997e-45 or 1.90000000000000003e-62 < t

   1. Initial program 60.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 60.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 62.8%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 61.5%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 61.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 61.3%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 61.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 61.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 61.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 61.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 61.3%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 61.3%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around 0 50.9%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 50.9%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 50.9%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. unpow2 50.9%

         \[\leadsto \frac{\ell \cdot \ell}{{t}^{3} \cdot \color{blue}{\left(k \cdot k\right)}} \]

   6. Simplified 50.9%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{{t}^{3} \cdot \left(k \cdot k\right)}} \]

   7. Taylor expanded in t around 0 50.9%

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]
   8. Step-by-step derivation

      1. unpow2 50.9%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      2. associate-*l* 59.6%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

   9. Simplified 59.6%

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

   if -5.3999999999999997e-45 < t < 1.90000000000000003e-62

   1. Initial program 34.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 34.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 34.3%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 34.2%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 34.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 34.2%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 34.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 34.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 34.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 34.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 34.2%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 34.2%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 65.7%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 65.7%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 65.6%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 65.6%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 65.6%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 87.0%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   6. Simplified 87.0%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

   7. Taylor expanded in l around 0 65.7%

      \[\leadsto 2 \cdot \color{blue}{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* 66.0%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{{\sin k}^{2} \cdot t}} \]

      2. associate-*r/ 65.9%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot \frac{{\ell}^{2}}{{k}^{2}}}}{{\sin k}^{2} \cdot t} \]

      3. unpow2 65.9%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{\sin k}^{2} \cdot t} \]

      4. unpow2 65.9%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{{\sin k}^{2} \cdot t} \]

      5. times-frac 87.3%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}}{{\sin k}^{2} \cdot t} \]

      6. unpow2 87.3%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}{{\sin k}^{2} \cdot t} \]

      7. *-commutative 87.3%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}}{{\sin k}^{2} \cdot t} \]

      8. *-commutative 87.3%

         \[\leadsto 2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}{\color{blue}{t \cdot {\sin k}^{2}}} \]

      9. times-frac 88.0%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   9. Simplified 88.0%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   10. Taylor expanded in k around 0 64.4%

       \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \color{blue}{\left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)}\right) \]
   11. Step-by-step derivation

       1. sub-neg 64.4%

          \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \color{blue}{\left(\frac{1}{{k}^{2}} + \left(-0.16666666666666666\right)\right)}\right) \]

       2. unpow2 64.4%

          \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \left(\frac{1}{\color{blue}{k \cdot k}} + \left(-0.16666666666666666\right)\right)\right) \]

       3. metadata-eval 64.4%

          \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \left(\frac{1}{k \cdot k} + \color{blue}{-0.16666666666666666}\right)\right) \]

   12. Simplified 64.4%

       \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \color{blue}{\left(\frac{1}{k \cdot k} + -0.16666666666666666\right)}\right) \]
   13. Step-by-step derivation

       1. pow2 64.4%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right) \]

   14. Applied egg-rr 64.4%

       \[\leadsto 2 \cdot \left(\frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.4 \cdot 10^{-45} \lor \neg \left(t \leq 1.9 \cdot 10^{-62}\right):\\ \;\;\;\;\frac{\ell \cdot \ell}{k \cdot \left({t}^{3} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right)\\ \end{array} \]

Alternative 13: 41.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.92 \cdot 10^{-236}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \ell}{t} \cdot -0.058333333333333334\right)\\ \mathbf{elif}\;\ell \leq 3.7 \cdot 10^{+227}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 1.92e-236)
   (* 2.0 (* (/ (* l l) t) -0.058333333333333334))
   (if (<= l 3.7e+227)
     (* 2.0 (* (/ l t) (/ l (pow k 4.0))))
     (*
      2.0
      (*
       (/ (* (/ l k) (/ l k)) t)
       (+ (/ 1.0 (* k k)) -0.16666666666666666))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 1.92e-236) {
		tmp = 2.0 * (((l * l) / t) * -0.058333333333333334);
	} else if (l <= 3.7e+227) {
		tmp = 2.0 * ((l / t) * (l / pow(k, 4.0)));
	} else {
		tmp = 2.0 * ((((l / k) * (l / k)) / t) * ((1.0 / (k * k)) + -0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 1.92d-236) then
        tmp = 2.0d0 * (((l * l) / t) * (-0.058333333333333334d0))
    else if (l <= 3.7d+227) then
        tmp = 2.0d0 * ((l / t) * (l / (k ** 4.0d0)))
    else
        tmp = 2.0d0 * ((((l / k) * (l / k)) / t) * ((1.0d0 / (k * k)) + (-0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 1.92e-236) {
		tmp = 2.0 * (((l * l) / t) * -0.058333333333333334);
	} else if (l <= 3.7e+227) {
		tmp = 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
	} else {
		tmp = 2.0 * ((((l / k) * (l / k)) / t) * ((1.0 / (k * k)) + -0.16666666666666666));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 1.92e-236:
		tmp = 2.0 * (((l * l) / t) * -0.058333333333333334)
	elif l <= 3.7e+227:
		tmp = 2.0 * ((l / t) * (l / math.pow(k, 4.0)))
	else:
		tmp = 2.0 * ((((l / k) * (l / k)) / t) * ((1.0 / (k * k)) + -0.16666666666666666))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 1.92e-236)
		tmp = Float64(2.0 * Float64(Float64(Float64(l * l) / t) * -0.058333333333333334));
	elseif (l <= 3.7e+227)
		tmp = Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(l / k) * Float64(l / k)) / t) * Float64(Float64(1.0 / Float64(k * k)) + -0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 1.92e-236)
		tmp = 2.0 * (((l * l) / t) * -0.058333333333333334);
	elseif (l <= 3.7e+227)
		tmp = 2.0 * ((l / t) * (l / (k ^ 4.0)));
	else
		tmp = 2.0 * ((((l / k) * (l / k)) / t) * ((1.0 / (k * k)) + -0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 1.92e-236], N[(2.0 * N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] * -0.058333333333333334), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.7e+227], N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.92e-236

   1. Initial program 49.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 49.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 51.1%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 50.3%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 50.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 50.1%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 50.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 50.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 50.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 50.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 50.1%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 50.1%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 54.2%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 54.2%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 55.0%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 55.0%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 55.0%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 62.6%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   6. Simplified 62.6%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

   7. Taylor expanded in k around 0 34.5%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\left(-0.058333333333333334 \cdot \frac{{k}^{2}}{t} + \frac{1}{{k}^{2} \cdot t}\right) - 0.16666666666666666 \cdot \frac{1}{t}\right)}\right) \]
   8. Step-by-step derivation

      1. fma-def 34.5%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\color{blue}{\mathsf{fma}\left(-0.058333333333333334, \frac{{k}^{2}}{t}, \frac{1}{{k}^{2} \cdot t}\right)} - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

      2. unpow2 34.5%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\mathsf{fma}\left(-0.058333333333333334, \frac{\color{blue}{k \cdot k}}{t}, \frac{1}{{k}^{2} \cdot t}\right) - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

      3. unpow2 34.5%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\mathsf{fma}\left(-0.058333333333333334, \frac{k \cdot k}{t}, \frac{1}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right) - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

      4. associate-*r/ 34.5%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\mathsf{fma}\left(-0.058333333333333334, \frac{k \cdot k}{t}, \frac{1}{\left(k \cdot k\right) \cdot t}\right) - \color{blue}{\frac{0.16666666666666666 \cdot 1}{t}}\right)\right) \]

      5. metadata-eval 34.5%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\mathsf{fma}\left(-0.058333333333333334, \frac{k \cdot k}{t}, \frac{1}{\left(k \cdot k\right) \cdot t}\right) - \frac{\color{blue}{0.16666666666666666}}{t}\right)\right) \]

   9. Simplified 34.5%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(-0.058333333333333334, \frac{k \cdot k}{t}, \frac{1}{\left(k \cdot k\right) \cdot t}\right) - \frac{0.16666666666666666}{t}\right)}\right) \]

   10. Taylor expanded in k around inf 33.3%

       \[\leadsto 2 \cdot \color{blue}{\left(-0.058333333333333334 \cdot \frac{{\ell}^{2}}{t}\right)} \]
   11. Step-by-step derivation

       1. *-commutative 33.3%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot -0.058333333333333334\right)} \]

       2. unpow2 33.3%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{t} \cdot -0.058333333333333334\right) \]

   12. Simplified 33.3%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell \cdot \ell}{t} \cdot -0.058333333333333334\right)} \]

   if 1.92e-236 < l < 3.6999999999999999e227

   1. Initial program 53.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 53.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 54.7%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 53.7%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 53.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 53.7%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 53.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 53.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 53.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 53.7%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 53.7%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 53.7%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 55.7%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 55.7%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 55.5%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 55.5%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 55.5%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 68.5%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   6. Simplified 68.5%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

   7. Taylor expanded in k around 0 47.7%

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. unpow2 47.7%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. *-commutative 47.7%

         \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

      3. times-frac 54.2%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   9. Simplified 54.2%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   if 3.6999999999999999e227 < l

   1. Initial program 27.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 27.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 27.1%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 27.1%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 27.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 27.1%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 27.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 27.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 27.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 27.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 27.1%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 27.1%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 47.7%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 47.7%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 47.7%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 47.7%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 47.7%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 80.1%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   6. Simplified 80.1%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

   7. Taylor expanded in l around 0 47.7%

      \[\leadsto 2 \cdot \color{blue}{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* 47.7%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{{\sin k}^{2} \cdot t}} \]

      2. associate-*r/ 47.7%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot \frac{{\ell}^{2}}{{k}^{2}}}}{{\sin k}^{2} \cdot t} \]

      3. unpow2 47.7%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{\sin k}^{2} \cdot t} \]

      4. unpow2 47.7%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{{\sin k}^{2} \cdot t} \]

      5. times-frac 80.1%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}}{{\sin k}^{2} \cdot t} \]

      6. unpow2 80.1%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}{{\sin k}^{2} \cdot t} \]

      7. *-commutative 80.1%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}}{{\sin k}^{2} \cdot t} \]

      8. *-commutative 80.1%

         \[\leadsto 2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}{\color{blue}{t \cdot {\sin k}^{2}}} \]

      9. times-frac 80.4%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   9. Simplified 80.4%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   10. Taylor expanded in k around 0 41.8%

       \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \color{blue}{\left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)}\right) \]
   11. Step-by-step derivation

       1. sub-neg 41.8%

          \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \color{blue}{\left(\frac{1}{{k}^{2}} + \left(-0.16666666666666666\right)\right)}\right) \]

       2. unpow2 41.8%

          \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \left(\frac{1}{\color{blue}{k \cdot k}} + \left(-0.16666666666666666\right)\right)\right) \]

       3. metadata-eval 41.8%

          \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \left(\frac{1}{k \cdot k} + \color{blue}{-0.16666666666666666}\right)\right) \]

   12. Simplified 41.8%

       \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \color{blue}{\left(\frac{1}{k \cdot k} + -0.16666666666666666\right)}\right) \]
   13. Step-by-step derivation

       1. pow2 41.8%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right) \]

   14. Applied egg-rr 41.8%

       \[\leadsto 2 \cdot \left(\frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.92 \cdot 10^{-236}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \ell}{t} \cdot -0.058333333333333334\right)\\ \mathbf{elif}\;\ell \leq 3.7 \cdot 10^{+227}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right)\\ \end{array} \]

Alternative 14: 41.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.4 \cdot 10^{-234}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \ell}{t} \cdot -0.058333333333333334\right)\\ \mathbf{elif}\;\ell \leq 9.6 \cdot 10^{+228}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 4.4e-234)
   (* 2.0 (* (/ (* l l) t) -0.058333333333333334))
   (if (<= l 9.6e+228)
     (* 2.0 (/ (/ l (/ t l)) (pow k 4.0)))
     (*
      2.0
      (*
       (/ (* (/ l k) (/ l k)) t)
       (+ (/ 1.0 (* k k)) -0.16666666666666666))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 4.4e-234) {
		tmp = 2.0 * (((l * l) / t) * -0.058333333333333334);
	} else if (l <= 9.6e+228) {
		tmp = 2.0 * ((l / (t / l)) / pow(k, 4.0));
	} else {
		tmp = 2.0 * ((((l / k) * (l / k)) / t) * ((1.0 / (k * k)) + -0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 4.4d-234) then
        tmp = 2.0d0 * (((l * l) / t) * (-0.058333333333333334d0))
    else if (l <= 9.6d+228) then
        tmp = 2.0d0 * ((l / (t / l)) / (k ** 4.0d0))
    else
        tmp = 2.0d0 * ((((l / k) * (l / k)) / t) * ((1.0d0 / (k * k)) + (-0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 4.4e-234) {
		tmp = 2.0 * (((l * l) / t) * -0.058333333333333334);
	} else if (l <= 9.6e+228) {
		tmp = 2.0 * ((l / (t / l)) / Math.pow(k, 4.0));
	} else {
		tmp = 2.0 * ((((l / k) * (l / k)) / t) * ((1.0 / (k * k)) + -0.16666666666666666));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 4.4e-234:
		tmp = 2.0 * (((l * l) / t) * -0.058333333333333334)
	elif l <= 9.6e+228:
		tmp = 2.0 * ((l / (t / l)) / math.pow(k, 4.0))
	else:
		tmp = 2.0 * ((((l / k) * (l / k)) / t) * ((1.0 / (k * k)) + -0.16666666666666666))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 4.4e-234)
		tmp = Float64(2.0 * Float64(Float64(Float64(l * l) / t) * -0.058333333333333334));
	elseif (l <= 9.6e+228)
		tmp = Float64(2.0 * Float64(Float64(l / Float64(t / l)) / (k ^ 4.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(l / k) * Float64(l / k)) / t) * Float64(Float64(1.0 / Float64(k * k)) + -0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 4.4e-234)
		tmp = 2.0 * (((l * l) / t) * -0.058333333333333334);
	elseif (l <= 9.6e+228)
		tmp = 2.0 * ((l / (t / l)) / (k ^ 4.0));
	else
		tmp = 2.0 * ((((l / k) * (l / k)) / t) * ((1.0 / (k * k)) + -0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 4.4e-234], N[(2.0 * N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] * -0.058333333333333334), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 9.6e+228], N[(2.0 * N[(N[(l / N[(t / l), $MachinePrecision]), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 4.3999999999999998e-234

   1. Initial program 49.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 49.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 51.1%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 50.3%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 50.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 50.1%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 50.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 50.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 50.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 50.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 50.1%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 50.1%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 54.2%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 54.2%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 55.0%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 55.0%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 55.0%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 62.4%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   6. Simplified 62.4%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

   7. Taylor expanded in k around 0 34.1%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\left(-0.058333333333333334 \cdot \frac{{k}^{2}}{t} + \frac{1}{{k}^{2} \cdot t}\right) - 0.16666666666666666 \cdot \frac{1}{t}\right)}\right) \]
   8. Step-by-step derivation

      1. fma-def 34.1%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\color{blue}{\mathsf{fma}\left(-0.058333333333333334, \frac{{k}^{2}}{t}, \frac{1}{{k}^{2} \cdot t}\right)} - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

      2. unpow2 34.1%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\mathsf{fma}\left(-0.058333333333333334, \frac{\color{blue}{k \cdot k}}{t}, \frac{1}{{k}^{2} \cdot t}\right) - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

      3. unpow2 34.1%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\mathsf{fma}\left(-0.058333333333333334, \frac{k \cdot k}{t}, \frac{1}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right) - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

      4. associate-*r/ 34.1%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\mathsf{fma}\left(-0.058333333333333334, \frac{k \cdot k}{t}, \frac{1}{\left(k \cdot k\right) \cdot t}\right) - \color{blue}{\frac{0.16666666666666666 \cdot 1}{t}}\right)\right) \]

      5. metadata-eval 34.1%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\mathsf{fma}\left(-0.058333333333333334, \frac{k \cdot k}{t}, \frac{1}{\left(k \cdot k\right) \cdot t}\right) - \frac{\color{blue}{0.16666666666666666}}{t}\right)\right) \]

   9. Simplified 34.1%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(-0.058333333333333334, \frac{k \cdot k}{t}, \frac{1}{\left(k \cdot k\right) \cdot t}\right) - \frac{0.16666666666666666}{t}\right)}\right) \]

   10. Taylor expanded in k around inf 33.5%

       \[\leadsto 2 \cdot \color{blue}{\left(-0.058333333333333334 \cdot \frac{{\ell}^{2}}{t}\right)} \]
   11. Step-by-step derivation

       1. *-commutative 33.5%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot -0.058333333333333334\right)} \]

       2. unpow2 33.5%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{t} \cdot -0.058333333333333334\right) \]

   12. Simplified 33.5%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell \cdot \ell}{t} \cdot -0.058333333333333334\right)} \]

   if 4.3999999999999998e-234 < l < 9.59999999999999954e228

   1. Initial program 53.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 53.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 54.8%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 53.8%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 53.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 53.8%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 53.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 53.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 53.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 53.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 53.8%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 53.8%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 55.8%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 55.8%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 55.6%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 55.6%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 55.6%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 68.9%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   6. Simplified 68.9%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

   7. Taylor expanded in l around 0 55.8%

      \[\leadsto 2 \cdot \color{blue}{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* 55.6%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{{\sin k}^{2} \cdot t}} \]

      2. associate-*r/ 55.6%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot \frac{{\ell}^{2}}{{k}^{2}}}}{{\sin k}^{2} \cdot t} \]

      3. unpow2 55.6%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{\sin k}^{2} \cdot t} \]

      4. unpow2 55.6%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{{\sin k}^{2} \cdot t} \]

      5. times-frac 68.8%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}}{{\sin k}^{2} \cdot t} \]

      6. unpow2 68.8%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}{{\sin k}^{2} \cdot t} \]

      7. *-commutative 68.8%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}}{{\sin k}^{2} \cdot t} \]

      8. *-commutative 68.8%

         \[\leadsto 2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}{\color{blue}{t \cdot {\sin k}^{2}}} \]

      9. times-frac 68.9%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   9. Simplified 68.9%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   10. Taylor expanded in k around 0 47.7%

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   11. Step-by-step derivation

       1. *-commutative 47.7%

          \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} \]

       2. associate-/r* 49.7%

          \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]

       3. unpow2 49.7%

          \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\ell \cdot \ell}}{t}}{{k}^{4}} \]

       4. associate-/l* 55.2%

          \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\ell}{\frac{t}{\ell}}}}{{k}^{4}} \]

   12. Simplified 55.2%

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}}} \]

   if 9.59999999999999954e228 < l

   1. Initial program 27.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 27.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 27.1%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 27.1%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 27.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 27.1%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 27.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 27.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 27.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 27.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 27.1%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 27.1%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 47.7%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 47.7%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 47.7%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 47.7%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 47.7%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 80.1%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   6. Simplified 80.1%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

   7. Taylor expanded in l around 0 47.7%

      \[\leadsto 2 \cdot \color{blue}{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* 47.7%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{{\sin k}^{2} \cdot t}} \]

      2. associate-*r/ 47.7%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot \frac{{\ell}^{2}}{{k}^{2}}}}{{\sin k}^{2} \cdot t} \]

      3. unpow2 47.7%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{\sin k}^{2} \cdot t} \]

      4. unpow2 47.7%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{{\sin k}^{2} \cdot t} \]

      5. times-frac 80.1%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}}{{\sin k}^{2} \cdot t} \]

      6. unpow2 80.1%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}{{\sin k}^{2} \cdot t} \]

      7. *-commutative 80.1%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}}{{\sin k}^{2} \cdot t} \]

      8. *-commutative 80.1%

         \[\leadsto 2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}{\color{blue}{t \cdot {\sin k}^{2}}} \]

      9. times-frac 80.4%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   9. Simplified 80.4%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   10. Taylor expanded in k around 0 41.8%

       \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \color{blue}{\left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)}\right) \]
   11. Step-by-step derivation

       1. sub-neg 41.8%

          \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \color{blue}{\left(\frac{1}{{k}^{2}} + \left(-0.16666666666666666\right)\right)}\right) \]

       2. unpow2 41.8%

          \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \left(\frac{1}{\color{blue}{k \cdot k}} + \left(-0.16666666666666666\right)\right)\right) \]

       3. metadata-eval 41.8%

          \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \left(\frac{1}{k \cdot k} + \color{blue}{-0.16666666666666666}\right)\right) \]

   12. Simplified 41.8%

       \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \color{blue}{\left(\frac{1}{k \cdot k} + -0.16666666666666666\right)}\right) \]
   13. Step-by-step derivation

       1. pow2 41.8%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right) \]

   14. Applied egg-rr 41.8%

       \[\leadsto 2 \cdot \left(\frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.4 \cdot 10^{-234}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \ell}{t} \cdot -0.058333333333333334\right)\\ \mathbf{elif}\;\ell \leq 9.6 \cdot 10^{+228}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right)\\ \end{array} \]

Alternative 15: 42.0% accurate, 20.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.55 \cdot 10^{-236}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \ell}{t} \cdot -0.058333333333333334\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 1.55e-236)
   (* 2.0 (* (/ (* l l) t) -0.058333333333333334))
   (*
    2.0
    (* (/ (* (/ l k) (/ l k)) t) (+ (/ 1.0 (* k k)) -0.16666666666666666)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 1.55e-236) {
		tmp = 2.0 * (((l * l) / t) * -0.058333333333333334);
	} else {
		tmp = 2.0 * ((((l / k) * (l / k)) / t) * ((1.0 / (k * k)) + -0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 1.55d-236) then
        tmp = 2.0d0 * (((l * l) / t) * (-0.058333333333333334d0))
    else
        tmp = 2.0d0 * ((((l / k) * (l / k)) / t) * ((1.0d0 / (k * k)) + (-0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 1.55e-236) {
		tmp = 2.0 * (((l * l) / t) * -0.058333333333333334);
	} else {
		tmp = 2.0 * ((((l / k) * (l / k)) / t) * ((1.0 / (k * k)) + -0.16666666666666666));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 1.55e-236:
		tmp = 2.0 * (((l * l) / t) * -0.058333333333333334)
	else:
		tmp = 2.0 * ((((l / k) * (l / k)) / t) * ((1.0 / (k * k)) + -0.16666666666666666))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 1.55e-236)
		tmp = Float64(2.0 * Float64(Float64(Float64(l * l) / t) * -0.058333333333333334));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(l / k) * Float64(l / k)) / t) * Float64(Float64(1.0 / Float64(k * k)) + -0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 1.55e-236)
		tmp = 2.0 * (((l * l) / t) * -0.058333333333333334);
	else
		tmp = 2.0 * ((((l / k) * (l / k)) / t) * ((1.0 / (k * k)) + -0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 1.55e-236], N[(2.0 * N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] * -0.058333333333333334), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.5499999999999999e-236

   1. Initial program 49.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 49.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 51.1%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 50.3%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 50.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 50.1%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 50.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 50.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 50.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 50.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 50.1%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 50.1%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 54.2%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 54.2%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 55.0%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 55.0%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 55.0%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 62.6%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   6. Simplified 62.6%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

   7. Taylor expanded in k around 0 34.5%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\left(-0.058333333333333334 \cdot \frac{{k}^{2}}{t} + \frac{1}{{k}^{2} \cdot t}\right) - 0.16666666666666666 \cdot \frac{1}{t}\right)}\right) \]
   8. Step-by-step derivation

      1. fma-def 34.5%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\color{blue}{\mathsf{fma}\left(-0.058333333333333334, \frac{{k}^{2}}{t}, \frac{1}{{k}^{2} \cdot t}\right)} - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

      2. unpow2 34.5%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\mathsf{fma}\left(-0.058333333333333334, \frac{\color{blue}{k \cdot k}}{t}, \frac{1}{{k}^{2} \cdot t}\right) - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

      3. unpow2 34.5%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\mathsf{fma}\left(-0.058333333333333334, \frac{k \cdot k}{t}, \frac{1}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right) - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

      4. associate-*r/ 34.5%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\mathsf{fma}\left(-0.058333333333333334, \frac{k \cdot k}{t}, \frac{1}{\left(k \cdot k\right) \cdot t}\right) - \color{blue}{\frac{0.16666666666666666 \cdot 1}{t}}\right)\right) \]

      5. metadata-eval 34.5%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\mathsf{fma}\left(-0.058333333333333334, \frac{k \cdot k}{t}, \frac{1}{\left(k \cdot k\right) \cdot t}\right) - \frac{\color{blue}{0.16666666666666666}}{t}\right)\right) \]

   9. Simplified 34.5%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(-0.058333333333333334, \frac{k \cdot k}{t}, \frac{1}{\left(k \cdot k\right) \cdot t}\right) - \frac{0.16666666666666666}{t}\right)}\right) \]

   10. Taylor expanded in k around inf 33.3%

       \[\leadsto 2 \cdot \color{blue}{\left(-0.058333333333333334 \cdot \frac{{\ell}^{2}}{t}\right)} \]
   11. Step-by-step derivation

       1. *-commutative 33.3%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot -0.058333333333333334\right)} \]

       2. unpow2 33.3%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{t} \cdot -0.058333333333333334\right) \]

   12. Simplified 33.3%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell \cdot \ell}{t} \cdot -0.058333333333333334\right)} \]

   if 1.5499999999999999e-236 < l

   1. Initial program 50.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 50.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 50.9%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 50.0%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 50.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 50.0%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 50.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 50.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 50.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 50.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 50.0%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 50.0%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 54.6%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 54.6%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 54.4%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 54.4%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 54.4%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 70.1%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   6. Simplified 70.1%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

   7. Taylor expanded in l around 0 54.6%

      \[\leadsto 2 \cdot \color{blue}{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* 54.4%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{{\sin k}^{2} \cdot t}} \]

      2. associate-*r/ 54.4%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot \frac{{\ell}^{2}}{{k}^{2}}}}{{\sin k}^{2} \cdot t} \]

      3. unpow2 54.4%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{\sin k}^{2} \cdot t} \]

      4. unpow2 54.4%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{{\sin k}^{2} \cdot t} \]

      5. times-frac 70.1%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}}{{\sin k}^{2} \cdot t} \]

      6. unpow2 70.1%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}{{\sin k}^{2} \cdot t} \]

      7. *-commutative 70.1%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}}{{\sin k}^{2} \cdot t} \]

      8. *-commutative 70.1%

         \[\leadsto 2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}{\color{blue}{t \cdot {\sin k}^{2}}} \]

      9. times-frac 70.2%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   9. Simplified 70.2%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   10. Taylor expanded in k around 0 49.4%

       \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \color{blue}{\left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)}\right) \]
   11. Step-by-step derivation

       1. sub-neg 49.4%

          \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \color{blue}{\left(\frac{1}{{k}^{2}} + \left(-0.16666666666666666\right)\right)}\right) \]

       2. unpow2 49.4%

          \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \left(\frac{1}{\color{blue}{k \cdot k}} + \left(-0.16666666666666666\right)\right)\right) \]

       3. metadata-eval 49.4%

          \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \left(\frac{1}{k \cdot k} + \color{blue}{-0.16666666666666666}\right)\right) \]

   12. Simplified 49.4%

       \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \color{blue}{\left(\frac{1}{k \cdot k} + -0.16666666666666666\right)}\right) \]
   13. Step-by-step derivation

       1. pow2 49.4%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right) \]

   14. Applied egg-rr 49.4%

       \[\leadsto 2 \cdot \left(\frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.55 \cdot 10^{-236}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \ell}{t} \cdot -0.058333333333333334\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right)\\ \end{array} \]

Alternative 16: 41.3% accurate, 22.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.1 \cdot 10^{-236}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \ell}{t} \cdot -0.058333333333333334\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{k \cdot \left(t \cdot k\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 2.1e-236)
   (* 2.0 (* (/ (* l l) t) -0.058333333333333334))
   (* 2.0 (* (* (/ l k) (/ l k)) (/ 1.0 (* k (* t k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 2.1e-236) {
		tmp = 2.0 * (((l * l) / t) * -0.058333333333333334);
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (1.0 / (k * (t * k))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 2.1d-236) then
        tmp = 2.0d0 * (((l * l) / t) * (-0.058333333333333334d0))
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) * (1.0d0 / (k * (t * k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 2.1e-236) {
		tmp = 2.0 * (((l * l) / t) * -0.058333333333333334);
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (1.0 / (k * (t * k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 2.1e-236:
		tmp = 2.0 * (((l * l) / t) * -0.058333333333333334)
	else:
		tmp = 2.0 * (((l / k) * (l / k)) * (1.0 / (k * (t * k))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 2.1e-236)
		tmp = Float64(2.0 * Float64(Float64(Float64(l * l) / t) * -0.058333333333333334));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(1.0 / Float64(k * Float64(t * k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 2.1e-236)
		tmp = 2.0 * (((l * l) / t) * -0.058333333333333334);
	else
		tmp = 2.0 * (((l / k) * (l / k)) * (1.0 / (k * (t * k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 2.1e-236], N[(2.0 * N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] * -0.058333333333333334), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.09999999999999979e-236

   1. Initial program 49.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 49.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 51.1%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 50.3%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 50.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 50.1%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 50.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 50.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 50.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 50.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 50.1%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 50.1%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 54.2%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 54.2%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 55.0%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 55.0%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 55.0%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 62.6%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   6. Simplified 62.6%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

   7. Taylor expanded in k around 0 34.5%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\left(-0.058333333333333334 \cdot \frac{{k}^{2}}{t} + \frac{1}{{k}^{2} \cdot t}\right) - 0.16666666666666666 \cdot \frac{1}{t}\right)}\right) \]
   8. Step-by-step derivation

      1. fma-def 34.5%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\color{blue}{\mathsf{fma}\left(-0.058333333333333334, \frac{{k}^{2}}{t}, \frac{1}{{k}^{2} \cdot t}\right)} - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

      2. unpow2 34.5%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\mathsf{fma}\left(-0.058333333333333334, \frac{\color{blue}{k \cdot k}}{t}, \frac{1}{{k}^{2} \cdot t}\right) - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

      3. unpow2 34.5%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\mathsf{fma}\left(-0.058333333333333334, \frac{k \cdot k}{t}, \frac{1}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right) - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

      4. associate-*r/ 34.5%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\mathsf{fma}\left(-0.058333333333333334, \frac{k \cdot k}{t}, \frac{1}{\left(k \cdot k\right) \cdot t}\right) - \color{blue}{\frac{0.16666666666666666 \cdot 1}{t}}\right)\right) \]

      5. metadata-eval 34.5%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\mathsf{fma}\left(-0.058333333333333334, \frac{k \cdot k}{t}, \frac{1}{\left(k \cdot k\right) \cdot t}\right) - \frac{\color{blue}{0.16666666666666666}}{t}\right)\right) \]

   9. Simplified 34.5%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(-0.058333333333333334, \frac{k \cdot k}{t}, \frac{1}{\left(k \cdot k\right) \cdot t}\right) - \frac{0.16666666666666666}{t}\right)}\right) \]

   10. Taylor expanded in k around inf 33.3%

       \[\leadsto 2 \cdot \color{blue}{\left(-0.058333333333333334 \cdot \frac{{\ell}^{2}}{t}\right)} \]
   11. Step-by-step derivation

       1. *-commutative 33.3%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot -0.058333333333333334\right)} \]

       2. unpow2 33.3%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{t} \cdot -0.058333333333333334\right) \]

   12. Simplified 33.3%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell \cdot \ell}{t} \cdot -0.058333333333333334\right)} \]

   if 2.09999999999999979e-236 < l

   1. Initial program 50.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 50.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 50.9%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 50.0%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 50.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 50.0%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 50.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 50.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 50.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 50.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 50.0%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 50.0%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 54.6%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 54.6%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 54.4%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 54.4%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 54.4%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 70.1%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   6. Simplified 70.1%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

   7. Taylor expanded in k around 0 51.7%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{1}{{k}^{2} \cdot t}}\right) \]
   8. Step-by-step derivation

      1. unpow2 51.7%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right) \]

      2. associate-*l* 51.7%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{\color{blue}{k \cdot \left(k \cdot t\right)}}\right) \]

   9. Simplified 51.7%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{1}{k \cdot \left(k \cdot t\right)}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.1 \cdot 10^{-236}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \ell}{t} \cdot -0.058333333333333334\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{k \cdot \left(t \cdot k\right)}\right)\\ \end{array} \]

Alternative 17: 41.3% accurate, 22.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.32 \cdot 10^{-236}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \ell}{t} \cdot -0.058333333333333334\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{t \cdot \left(k \cdot k\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 1.32e-236)
   (* 2.0 (* (/ (* l l) t) -0.058333333333333334))
   (* 2.0 (* (* (/ l k) (/ l k)) (/ 1.0 (* t (* k k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 1.32e-236) {
		tmp = 2.0 * (((l * l) / t) * -0.058333333333333334);
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (1.0 / (t * (k * k))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 1.32d-236) then
        tmp = 2.0d0 * (((l * l) / t) * (-0.058333333333333334d0))
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) * (1.0d0 / (t * (k * k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 1.32e-236) {
		tmp = 2.0 * (((l * l) / t) * -0.058333333333333334);
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (1.0 / (t * (k * k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 1.32e-236:
		tmp = 2.0 * (((l * l) / t) * -0.058333333333333334)
	else:
		tmp = 2.0 * (((l / k) * (l / k)) * (1.0 / (t * (k * k))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 1.32e-236)
		tmp = Float64(2.0 * Float64(Float64(Float64(l * l) / t) * -0.058333333333333334));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(1.0 / Float64(t * Float64(k * k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 1.32e-236)
		tmp = 2.0 * (((l * l) / t) * -0.058333333333333334);
	else
		tmp = 2.0 * (((l / k) * (l / k)) * (1.0 / (t * (k * k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 1.32e-236], N[(2.0 * N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] * -0.058333333333333334), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.32e-236

   1. Initial program 49.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 49.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 51.1%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 50.3%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 50.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 50.1%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 50.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 50.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 50.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 50.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 50.1%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 50.1%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 54.2%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 54.2%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 55.0%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 55.0%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 55.0%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 62.6%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   6. Simplified 62.6%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

   7. Taylor expanded in k around 0 34.5%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\left(-0.058333333333333334 \cdot \frac{{k}^{2}}{t} + \frac{1}{{k}^{2} \cdot t}\right) - 0.16666666666666666 \cdot \frac{1}{t}\right)}\right) \]
   8. Step-by-step derivation

      1. fma-def 34.5%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\color{blue}{\mathsf{fma}\left(-0.058333333333333334, \frac{{k}^{2}}{t}, \frac{1}{{k}^{2} \cdot t}\right)} - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

      2. unpow2 34.5%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\mathsf{fma}\left(-0.058333333333333334, \frac{\color{blue}{k \cdot k}}{t}, \frac{1}{{k}^{2} \cdot t}\right) - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

      3. unpow2 34.5%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\mathsf{fma}\left(-0.058333333333333334, \frac{k \cdot k}{t}, \frac{1}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right) - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

      4. associate-*r/ 34.5%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\mathsf{fma}\left(-0.058333333333333334, \frac{k \cdot k}{t}, \frac{1}{\left(k \cdot k\right) \cdot t}\right) - \color{blue}{\frac{0.16666666666666666 \cdot 1}{t}}\right)\right) \]

      5. metadata-eval 34.5%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\mathsf{fma}\left(-0.058333333333333334, \frac{k \cdot k}{t}, \frac{1}{\left(k \cdot k\right) \cdot t}\right) - \frac{\color{blue}{0.16666666666666666}}{t}\right)\right) \]

   9. Simplified 34.5%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(-0.058333333333333334, \frac{k \cdot k}{t}, \frac{1}{\left(k \cdot k\right) \cdot t}\right) - \frac{0.16666666666666666}{t}\right)}\right) \]

   10. Taylor expanded in k around inf 33.3%

       \[\leadsto 2 \cdot \color{blue}{\left(-0.058333333333333334 \cdot \frac{{\ell}^{2}}{t}\right)} \]
   11. Step-by-step derivation

       1. *-commutative 33.3%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot -0.058333333333333334\right)} \]

       2. unpow2 33.3%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{t} \cdot -0.058333333333333334\right) \]

   12. Simplified 33.3%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell \cdot \ell}{t} \cdot -0.058333333333333334\right)} \]

   if 1.32e-236 < l

   1. Initial program 50.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 50.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 50.9%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 50.0%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 50.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 50.0%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 50.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 50.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 50.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 50.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 50.0%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 50.0%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 54.6%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 54.6%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 54.4%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 54.4%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 54.4%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 70.1%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   6. Simplified 70.1%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

   7. Taylor expanded in k around 0 51.7%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{1}{{k}^{2} \cdot t}}\right) \]
   8. Step-by-step derivation

      1. unpow2 51.7%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right) \]

   9. Simplified 51.7%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{1}{\left(k \cdot k\right) \cdot t}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.32 \cdot 10^{-236}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \ell}{t} \cdot -0.058333333333333334\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{t \cdot \left(k \cdot k\right)}\right)\\ \end{array} \]

Alternative 18: 29.5% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 10000:\\ \;\;\;\;-0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(-0.16666666666666666 \cdot \frac{\ell \cdot \ell}{t \cdot \left(k \cdot k\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 10000.0)
   (* -0.11666666666666667 (* l (/ l t)))
   (* 2.0 (* -0.16666666666666666 (/ (* l l) (* t (* k k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 10000.0) {
		tmp = -0.11666666666666667 * (l * (l / t));
	} else {
		tmp = 2.0 * (-0.16666666666666666 * ((l * l) / (t * (k * k))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 10000.0d0) then
        tmp = (-0.11666666666666667d0) * (l * (l / t))
    else
        tmp = 2.0d0 * ((-0.16666666666666666d0) * ((l * l) / (t * (k * k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 10000.0) {
		tmp = -0.11666666666666667 * (l * (l / t));
	} else {
		tmp = 2.0 * (-0.16666666666666666 * ((l * l) / (t * (k * k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 10000.0:
		tmp = -0.11666666666666667 * (l * (l / t))
	else:
		tmp = 2.0 * (-0.16666666666666666 * ((l * l) / (t * (k * k))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 10000.0)
		tmp = Float64(-0.11666666666666667 * Float64(l * Float64(l / t)));
	else
		tmp = Float64(2.0 * Float64(-0.16666666666666666 * Float64(Float64(l * l) / Float64(t * Float64(k * k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 10000.0)
		tmp = -0.11666666666666667 * (l * (l / t));
	else
		tmp = 2.0 * (-0.16666666666666666 * ((l * l) / (t * (k * k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 10000.0], N[(-0.11666666666666667 * N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(-0.16666666666666666 * N[(N[(l * l), $MachinePrecision] / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1e4

   1. Initial program 51.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 51.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 52.8%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 51.7%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 51.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 51.6%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 51.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 51.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 51.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 51.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 51.6%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 51.6%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 48.2%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 48.2%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 49.4%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 49.4%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 49.4%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 58.0%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   6. Simplified 58.0%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

   7. Taylor expanded in k around 0 40.0%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\left(-0.058333333333333334 \cdot \frac{{k}^{2}}{t} + \frac{1}{{k}^{2} \cdot t}\right) - 0.16666666666666666 \cdot \frac{1}{t}\right)}\right) \]
   8. Step-by-step derivation

      1. fma-def 40.0%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\color{blue}{\mathsf{fma}\left(-0.058333333333333334, \frac{{k}^{2}}{t}, \frac{1}{{k}^{2} \cdot t}\right)} - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

      2. unpow2 40.0%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\mathsf{fma}\left(-0.058333333333333334, \frac{\color{blue}{k \cdot k}}{t}, \frac{1}{{k}^{2} \cdot t}\right) - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

      3. unpow2 40.0%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\mathsf{fma}\left(-0.058333333333333334, \frac{k \cdot k}{t}, \frac{1}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right) - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

      4. associate-*r/ 40.0%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\mathsf{fma}\left(-0.058333333333333334, \frac{k \cdot k}{t}, \frac{1}{\left(k \cdot k\right) \cdot t}\right) - \color{blue}{\frac{0.16666666666666666 \cdot 1}{t}}\right)\right) \]

      5. metadata-eval 40.0%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\mathsf{fma}\left(-0.058333333333333334, \frac{k \cdot k}{t}, \frac{1}{\left(k \cdot k\right) \cdot t}\right) - \frac{\color{blue}{0.16666666666666666}}{t}\right)\right) \]

   9. Simplified 40.0%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(-0.058333333333333334, \frac{k \cdot k}{t}, \frac{1}{\left(k \cdot k\right) \cdot t}\right) - \frac{0.16666666666666666}{t}\right)}\right) \]

   10. Taylor expanded in k around inf 21.4%

       \[\leadsto 2 \cdot \color{blue}{\left(-0.058333333333333334 \cdot \frac{{\ell}^{2}}{t}\right)} \]
   11. Step-by-step derivation

       1. *-commutative 21.4%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot -0.058333333333333334\right)} \]

       2. unpow2 21.4%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{t} \cdot -0.058333333333333334\right) \]

       3. associate-/l* 20.6%

          \[\leadsto 2 \cdot \left(\color{blue}{\frac{\ell}{\frac{t}{\ell}}} \cdot -0.058333333333333334\right) \]

   12. Simplified 20.6%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{\frac{t}{\ell}} \cdot -0.058333333333333334\right)} \]

   13. Taylor expanded in l around 0 21.4%

       \[\leadsto \color{blue}{-0.11666666666666667 \cdot \frac{{\ell}^{2}}{t}} \]
   14. Step-by-step derivation

       1. unpow2 21.4%

          \[\leadsto -0.11666666666666667 \cdot \frac{\color{blue}{\ell \cdot \ell}}{t} \]

       2. associate-*r/ 20.6%

          \[\leadsto -0.11666666666666667 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{t}\right)} \]

   15. Simplified 20.6%

       \[\leadsto \color{blue}{-0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{t}\right)} \]

   if 1e4 < k

   1. Initial program 45.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 45.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 45.9%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 45.9%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 45.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 45.9%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 45.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 45.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 45.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 45.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 45.9%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 45.9%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 71.3%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 71.3%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 69.6%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 69.6%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 69.6%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 87.1%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   6. Simplified 87.1%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

   7. Taylor expanded in l around 0 71.3%

      \[\leadsto 2 \cdot \color{blue}{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* 69.6%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{{\sin k}^{2} \cdot t}} \]

      2. associate-*r/ 69.6%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot \frac{{\ell}^{2}}{{k}^{2}}}}{{\sin k}^{2} \cdot t} \]

      3. unpow2 69.6%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{\sin k}^{2} \cdot t} \]

      4. unpow2 69.6%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{{\sin k}^{2} \cdot t} \]

      5. times-frac 87.0%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}}{{\sin k}^{2} \cdot t} \]

      6. unpow2 87.0%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}{{\sin k}^{2} \cdot t} \]

      7. *-commutative 87.0%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}}{{\sin k}^{2} \cdot t} \]

      8. *-commutative 87.0%

         \[\leadsto 2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}{\color{blue}{t \cdot {\sin k}^{2}}} \]

      9. times-frac 87.1%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   9. Simplified 87.1%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   10. Taylor expanded in k around 0 57.1%

       \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \color{blue}{\left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)}\right) \]
   11. Step-by-step derivation

       1. sub-neg 57.1%

          \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \color{blue}{\left(\frac{1}{{k}^{2}} + \left(-0.16666666666666666\right)\right)}\right) \]

       2. unpow2 57.1%

          \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \left(\frac{1}{\color{blue}{k \cdot k}} + \left(-0.16666666666666666\right)\right)\right) \]

       3. metadata-eval 57.1%

          \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \left(\frac{1}{k \cdot k} + \color{blue}{-0.16666666666666666}\right)\right) \]

   12. Simplified 57.1%

       \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \color{blue}{\left(\frac{1}{k \cdot k} + -0.16666666666666666\right)}\right) \]

   13. Taylor expanded in k around inf 56.5%

       \[\leadsto 2 \cdot \color{blue}{\left(-0.16666666666666666 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]
   14. Step-by-step derivation

       1. *-commutative 56.5%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot -0.16666666666666666\right)} \]

       2. unpow2 56.5%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t} \cdot -0.16666666666666666\right) \]

       3. unpow2 56.5%

          \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot -0.16666666666666666\right) \]

   15. Simplified 56.5%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.16666666666666666\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10000:\\ \;\;\;\;-0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(-0.16666666666666666 \cdot \frac{\ell \cdot \ell}{t \cdot \left(k \cdot k\right)}\right)\\ \end{array} \]

Alternative 19: 25.8% accurate, 46.8× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\frac{\ell \cdot \ell}{t} \cdot -0.058333333333333334\right) \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* 2.0 (* (/ (* l l) t) -0.058333333333333334)))

C

double code(double t, double l, double k) {
	return 2.0 * (((l * l) / t) * -0.058333333333333334);
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * (((l * l) / t) * (-0.058333333333333334d0))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * (((l * l) / t) * -0.058333333333333334);
}

Python

def code(t, l, k):
	return 2.0 * (((l * l) / t) * -0.058333333333333334)

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(Float64(l * l) / t) * -0.058333333333333334))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * (((l * l) / t) * -0.058333333333333334);
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] * -0.058333333333333334), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/l/ 49.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   2. associate-*l/ 51.0%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

   3. associate-*l/ 50.2%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

   4. associate-/r/ 50.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

   5. *-commutative 50.1%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

   6. associate-/l/ 50.1%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   7. associate-*r* 50.1%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   8. *-commutative 50.1%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

   9. associate-*r* 50.1%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

   10. *-commutative 50.1%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

3. Simplified 50.1%

   \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

4. Taylor expanded in k around inf 54.4%

   \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
5. Step-by-step derivation

   1. *-commutative 54.4%

      \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

   2. times-frac 54.8%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

   3. unpow2 54.8%

      \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   4. unpow2 54.8%

      \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   5. times-frac 65.7%

      \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

6. Simplified 65.7%

   \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

7. Taylor expanded in k around 0 33.3%

   \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\left(-0.058333333333333334 \cdot \frac{{k}^{2}}{t} + \frac{1}{{k}^{2} \cdot t}\right) - 0.16666666666666666 \cdot \frac{1}{t}\right)}\right) \]
8. Step-by-step derivation

   1. fma-def 33.3%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\color{blue}{\mathsf{fma}\left(-0.058333333333333334, \frac{{k}^{2}}{t}, \frac{1}{{k}^{2} \cdot t}\right)} - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

   2. unpow2 33.3%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\mathsf{fma}\left(-0.058333333333333334, \frac{\color{blue}{k \cdot k}}{t}, \frac{1}{{k}^{2} \cdot t}\right) - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

   3. unpow2 33.3%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\mathsf{fma}\left(-0.058333333333333334, \frac{k \cdot k}{t}, \frac{1}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right) - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

   4. associate-*r/ 33.3%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\mathsf{fma}\left(-0.058333333333333334, \frac{k \cdot k}{t}, \frac{1}{\left(k \cdot k\right) \cdot t}\right) - \color{blue}{\frac{0.16666666666666666 \cdot 1}{t}}\right)\right) \]

   5. metadata-eval 33.3%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\mathsf{fma}\left(-0.058333333333333334, \frac{k \cdot k}{t}, \frac{1}{\left(k \cdot k\right) \cdot t}\right) - \frac{\color{blue}{0.16666666666666666}}{t}\right)\right) \]

9. Simplified 33.3%

   \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(-0.058333333333333334, \frac{k \cdot k}{t}, \frac{1}{\left(k \cdot k\right) \cdot t}\right) - \frac{0.16666666666666666}{t}\right)}\right) \]

10. Taylor expanded in k around inf 26.7%

    \[\leadsto 2 \cdot \color{blue}{\left(-0.058333333333333334 \cdot \frac{{\ell}^{2}}{t}\right)} \]
11. Step-by-step derivation

    1. *-commutative 26.7%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot -0.058333333333333334\right)} \]

    2. unpow2 26.7%

       \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{t} \cdot -0.058333333333333334\right) \]

12. Simplified 26.7%

    \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell \cdot \ell}{t} \cdot -0.058333333333333334\right)} \]

13. Final simplification 26.7%

    \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{t} \cdot -0.058333333333333334\right) \]

Alternative 20: 23.4% accurate, 60.1× speedup

Math

\[\begin{array}{l} \\ -0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{t}\right) \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* -0.11666666666666667 (* l (/ l t))))

C

double code(double t, double l, double k) {
	return -0.11666666666666667 * (l * (l / t));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (-0.11666666666666667d0) * (l * (l / t))
end function

Java

public static double code(double t, double l, double k) {
	return -0.11666666666666667 * (l * (l / t));
}

Python

def code(t, l, k):
	return -0.11666666666666667 * (l * (l / t))

Julia

function code(t, l, k)
	return Float64(-0.11666666666666667 * Float64(l * Float64(l / t)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = -0.11666666666666667 * (l * (l / t));
end

Wolfram

code[t_, l_, k_] := N[(-0.11666666666666667 * N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/l/ 49.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   2. associate-*l/ 51.0%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

   3. associate-*l/ 50.2%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

   4. associate-/r/ 50.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

   5. *-commutative 50.1%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

   6. associate-/l/ 50.1%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   7. associate-*r* 50.1%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   8. *-commutative 50.1%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

   9. associate-*r* 50.1%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

   10. *-commutative 50.1%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

3. Simplified 50.1%

   \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

4. Taylor expanded in k around inf 54.4%

   \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
5. Step-by-step derivation

   1. *-commutative 54.4%

      \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

   2. times-frac 54.8%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

   3. unpow2 54.8%

      \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   4. unpow2 54.8%

      \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   5. times-frac 65.7%

      \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

6. Simplified 65.7%

   \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

7. Taylor expanded in k around 0 33.3%

   \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\left(-0.058333333333333334 \cdot \frac{{k}^{2}}{t} + \frac{1}{{k}^{2} \cdot t}\right) - 0.16666666666666666 \cdot \frac{1}{t}\right)}\right) \]
8. Step-by-step derivation

   1. fma-def 33.3%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\color{blue}{\mathsf{fma}\left(-0.058333333333333334, \frac{{k}^{2}}{t}, \frac{1}{{k}^{2} \cdot t}\right)} - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

   2. unpow2 33.3%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\mathsf{fma}\left(-0.058333333333333334, \frac{\color{blue}{k \cdot k}}{t}, \frac{1}{{k}^{2} \cdot t}\right) - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

   3. unpow2 33.3%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\mathsf{fma}\left(-0.058333333333333334, \frac{k \cdot k}{t}, \frac{1}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right) - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

   4. associate-*r/ 33.3%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\mathsf{fma}\left(-0.058333333333333334, \frac{k \cdot k}{t}, \frac{1}{\left(k \cdot k\right) \cdot t}\right) - \color{blue}{\frac{0.16666666666666666 \cdot 1}{t}}\right)\right) \]

   5. metadata-eval 33.3%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\mathsf{fma}\left(-0.058333333333333334, \frac{k \cdot k}{t}, \frac{1}{\left(k \cdot k\right) \cdot t}\right) - \frac{\color{blue}{0.16666666666666666}}{t}\right)\right) \]

9. Simplified 33.3%

   \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(-0.058333333333333334, \frac{k \cdot k}{t}, \frac{1}{\left(k \cdot k\right) \cdot t}\right) - \frac{0.16666666666666666}{t}\right)}\right) \]

10. Taylor expanded in k around inf 26.7%

    \[\leadsto 2 \cdot \color{blue}{\left(-0.058333333333333334 \cdot \frac{{\ell}^{2}}{t}\right)} \]
11. Step-by-step derivation

    1. *-commutative 26.7%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot -0.058333333333333334\right)} \]

    2. unpow2 26.7%

       \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{t} \cdot -0.058333333333333334\right) \]

    3. associate-/l* 24.0%

       \[\leadsto 2 \cdot \left(\color{blue}{\frac{\ell}{\frac{t}{\ell}}} \cdot -0.058333333333333334\right) \]

12. Simplified 24.0%

    \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{\frac{t}{\ell}} \cdot -0.058333333333333334\right)} \]

13. Taylor expanded in l around 0 26.7%

    \[\leadsto \color{blue}{-0.11666666666666667 \cdot \frac{{\ell}^{2}}{t}} \]
14. Step-by-step derivation

    1. unpow2 26.7%

       \[\leadsto -0.11666666666666667 \cdot \frac{\color{blue}{\ell \cdot \ell}}{t} \]

    2. associate-*r/ 24.0%

       \[\leadsto -0.11666666666666667 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{t}\right)} \]

15. Simplified 24.0%

    \[\leadsto \color{blue}{-0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{t}\right)} \]

16. Final simplification 24.0%

    \[\leadsto -0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{t}\right) \]

Reproduce

herbie shell --seed 2023207 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  :precision binary64
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))